UC-NRLF 


271    DbD 


AN  INTRODUCTION  TO  j 
LABORATORY       I 


T.UTTLE 


CO 

cr 
o 

DQ 


UNIVERSITY  OF  CALIFORNIA 


LIBRARY 

OF  THE 


Received .V»«... 

Accessions  No.     ff  Ufjf  Book  Afo..._/ 


GIFT  OF 
Author 


AN  INTRODUCTION 


TO 


LABORATORF-MYSICS 


BY 

LUCIUS   TUTTLE 

ASSOCIATE  IN  PHYSICS,  JEFFERSON  MEDICAL  COLLEGE,   PHILADELPHIA 


PHILADELPHIA 

JEFFERSON  LABORATORY  OF  PHYSICS 
1915 


COPYRIGHT  1915  BY  LUCIUS  TUTTLE 


PUBLISHED  BY  THE  AUTHOR 


PREFACE 


THIS  book  is  essentially  a  revision  of  the  mimeo- 
graphed direction  sheets  that  have  been  used  in  the 
first  part  of  the  laboratory  course  given  by  the  writer 
at  Jefferson  Medical  College.  It  does  not  cover  the 
ground  of  the  usual  laboratory  manuals  of  physics, 
but  is  intended  to  precede  the  use  of  any  one  of  them 
in  a  course  of  physical  measurement,  and  includes  the 
matter  that  is  usually  relegated  to  an  introductory 
chapter  or  an  appendix,  where  the  student  is  not  apt  to 
get  as  good  a  grasp  of  it  as  he  does  of  the  subjects  that 
are  emphasized  by  his  experimental  work.  In  addi- 
tion to  the  statements  of  facts  and  theory  each  of  the 
fifteen  lessons  in  the  book  includes  directions  for  actual 
experimental  work  to  be  performed  by  the  student, 
and  the  amount  of  this  work  has  been  so  planned  that 
each  lesson  will  require  the  same  length  of  time  as  any 
of  the  others.  For  the  average  student  this  means  about 
three  hours,  but  the  material  of  the  lessons  can  easily 
be  divided  into  a  greater  number  of  shorter  exercises  if 
desirable. 

Explanations  and  directions  have  been  given  with 
considerable  detail,  partly  in  order  to  avoid  the  neces- 
sity for  continuous  oral  assistance  on  the  part  of  the 
instructor,  and  partly  to  help  the  student  to  learn 
with  a  minimum  of  deliberate  memorizing.  For  the 
latter  purpose  facts  have  sometimes  been  stated 

673213 


iv  PREFACE 

implicitly  instead  of  explicitly,  and  later  have  been 
reiterated  in  a  more  expositional  form. 

At  first  glance  the  book  will  seem  more  mathematical 
than  it  really  is,  for  the  re-statement  of  some  ele- 
mentary principle  is  occasionally  helpful  to  any  student. 
No  knowledge  of  trigonometry,  however,  is  presupposed, 
and  none  is  imposed  upon  the  reader  of  the  book,  the 
terms  "function,"  " tangent,"  "  cosine,"  etc.,  that  will 
occasionally  be  found  being  used  merely  as  convenient 
abbreviations  for  ideas  that  would  otherwise  need  a 
more  cumbrous  description. 

In  the  introductory  chapter  the  commonest  mathe- 
matical deficiencies  of  the  student  are  reviewed  and 
an  opportunity  is  given  him  to  test  his  weak  points. 
A  lesson  on  logarithms  is  included,  which  can  be 
omitted,  if  preferred,  by  a  class  that  is  familiar  with 
the  subject;  but  there  are  often  members  of  such  a 
class  who  cannot  make  practical  use  of  logarithmic 
tables  readily,  or  even  accurately,  without  additional 
practice,  and  to  anyone  who  does  not  need  the  practice 
it  will  not  be  at  all  irksome.  Care  has  been  taken  to 
make  the  tables  both  accurate  and  convenient.  Exper- 
ience has  shown  that  the  somewhat  unconventional 
arrangement  of  the  table  of  probable  errors  (page  141) 
is  the  most  satisfactory  in  actual  use.  The  table  of 
logarithmic  circular  functions  has  been  given  the  greatest 
possible  compactness.  The  columns  of  the  table  of 
four-place  logarithms  are  arranged  especially  for  the 
convenience  of  the  student  of  physics  and  the  propor- 
tional parts  are  given  in  the  same  way  as  in  carefully 
constructed  larger  tables.  None  of  the  methods  of 
arranging  a  five-place  table  with  proportional  parts 


PREFACE  v 

within  the  limits  of  two  pages  has  ever  succeeded  in  giv- 
ing the  fifth  figure  satisfactorily,  and  several  books  for 
physicists  have  been  published  in  which  even  the  fourth 
figure  of  such  a  table  will  often  be  found  incorrect. 
Accordingly,  for  the  five-place  table  in  the  present  vol- 
ume no  attempt  has  been  made  to  include  propor- 
tional parts,  but  a  rule  has  been  given  that  will  enable 
the  interpolation  to  be  performed  mentally.  This  may 
seem  somewhat  troublesome,  at  first,  to  one  who  is  not 
used  to  logarithmic  computation,  but  after  a  little 
practice  it  will  be  found  to  present  no  difficulties.  As  a 
five-place  table  is  needed  only  occasionally  the  arrange- 
ment of  columns  as  in  the  four-place  table  has  been 
sacrificed  for  one  which  allows  the  tabular  differences 
to  be  given  at  more  regular  and  convenient  intervals. 

I  have  replaced  the  perpetually  misleading  com- 
mon name  for  the  representative  value  of  a  set  of 
residuals  by  one  which  is  free  from  this  objection 
and  at  the  same  time  suggests  the  nature  of  the  quan- 
tity in  question.  A  few  other  innovations  will  be  found 
scattered  through  the  text,  but  for  the  most  part  the 
book  follows  well-beaten  lines.  The  final  lesson  will 
probably  seem  harder  than  the  preceding  ones,  to  most 
students;  but,  if  desirable,  it  can  be  considered  as  a  sort 
of  appendix,  and  used  only  for  reference. 

I  have  found  it  advisable  to  devote  the  first  ten  or 
fifteen  minutes  of  the  laboratory  period  to  a  rapid 
recitation  based  on  the  lesson  of  the  previous  day; 
and  have  allowed  the  students  to  compare  many  of 
their  important  numerical  determinations  by  having 
them  record  certain  specified  results  each  day  upon 
a  large  card  (22|"  X  26|")  that  is  kept  on  one  of  the 


vi  PREFACE 

laboratory  tables,  and  is  ruled  into  separate  columns 
headed  by  each  student's  name  and  having  separate 
lines  for  each  datum.  For  the  fifteen  exercises  of  this 
book  the  following  data  may  be  suggested: 

Weights  and  measures:  density  of  a  (brass-and-air) 
weight. 

Angles :  relatively  largest  error  of  measured  sines. 

Accuracy :  experimental  value  of  TT,  and  its  error. 

Logarithms:  calculated  value  of  1  +  1  +  1/1.2  + 
1/1.2.3  +  1/1.2.3.4+  .... 

Small  magnitudes:  results  and  mean  of  a  double 
weighing. 

Slide  rule:  approximate  ratio  for  TT,  different  from  22/7. 

Graphic  method:  least  x  for  which  exp  (—  x2)  is  indis- 
tinguishable from  zero. 

Graphic  analysis:  equation  of  black-thread  experi- 
ment. 

Method  of  coincidence:  measured  length  of  an  inch; 
or  slide-rule  ratio  of  1  gm  to  1  grain. 

Measurements:  mode  and  extremes  of  measured 
variates. 

Statistics:  average,  median,  and  quartiles  of  variates. 

Dispersion:  comparison  of  semi-interquartile  range 
and  dispersion. 

Weights :  weighted  average  for  the  density  of  aluminum. 

Criteria  of  rejection:  closer  values  of  the  ratios 
10  :  12  :  15. 

Least  squares  and  errors :  displacement  of  the  second 
hand. 

Most  of  the  apparatus  required  will  be  found  to  be 
included  in  that  which  is  used  in  other  physical  experi- 


PREFACE  vii 

ments;  a  complete  list  of  what  is  needed  for  each 
group  of  two  students  is  given  here. 

2  metre  sticks  (graduated  in  tenths  of  an  inch  on 
the  back). 

2  30-cm  rulers. 

1  50-cm3  graduated  cylinder. 

1  10-cm3  graduated  pipette. 

1  platform  balance  or  trip  scale  with  slide  giving 
tenths  of  a  gram. 

1  set  of  brass  weights,  1  gm  to  500  gm. 

1  set  of  iron  weights,  1  oz  to  8  oz. 

1  pair  of  fine-pointed  dividers. 

1  combined  protractor  and  diagonal  scale. 

1  brass  disc  for  the  measurement  of  TT. 

1  ten-inch  slide  rule  without  celluloid  facings  but 
provided  with  A,  B,  C,  D,  S,  L,  T,  scales,  metric  equiva- 
lents, and  a  runner. 

1  hard  wood  block. 

1  vernier  caliper. 

100  seeds  or  other  variates. 

1  aluminum  block  for  density  measurements. 

1  set  of  ''overflow  can"  and  " catch-bucket"  for 
Archimedes'  Principle. 

2  square  wooden  rods  for  balance  pans. 

1  iron  clamp  to  hold  balance  on  cross-bar  over  table. 

String. 

Fine  black  thread. 

Cardboard. 

Large  wire  nail. 

Test-tube. 

«   The  student  should  have  a  watch  with  a  second  hand, 
a   pocket-knife,    and   the   supplies   mentioned   in   the 


viii  PREFACE 

introduction ;  a  clock  that  beats  audible  seconds  should 
be  available.  The  slide  rule  should  have  6745  on  the 
C  scale  marked  by  making  a  shallow  cut  with  a  sharp 
knife  and  rubbing  in  a  little  oil  pigment.  The  note- 
book used  at  the  Jefferson  Laboratory  of  Physics 
measures  about  eight  by  ten  and  a  half  inches  and  is 
ruled  both  horizontally  and  vertically  at  intervals  of 
one-seventh  of  an  inch. 


CONTENTS 


LESSON  PAGE 

I.  WEIGHTS  AND  MEASURES 20 

C.  G.  S.  System.  Units  of  Length.  Units  of 
Area  and  Volume.  Units  of  Mass  and  Density. 
Unit  of  Time.  Practice  in  Using  the  C.  G.  S. 
System.  Rule  for  Rounding  Off  a  Half.  The 
Hand  as  a  Measure.  Measurement  of  Area. 
Measurement  of  Volume.  Measurement  of 
Mass.  Measurement  of  Density. 

II.  ANGLES  AND  CIRCULAR  FUNCTIONS 27 

Unit  of  Angle.  Numerical  Measure  of  an  Angle. 
Use  of  the  Protractor.  The  Diagonal  Scale. 
The  Tangent  of  an  Angle.  Use  of  the  Table. 
The  Sine  of  an  Angle.  Definition  of  Function. 
The  Cosine  of  an  Angle. 

III.  ACCURACY  AND  SIGNIFICANT  FIGURES     ....       34 

Estimation  of  Tenths.  Mistakes.  The  Value  of 
TT.  Physical  Measurement.  Ideal  Accuracy. 
Decimal  Accuracy.  Significant  Figures.  Rel- 
ative Accuracy.  Calculation  of  Relative 
Errors.  Rule  for  Relative  Difference.  Stand- 
ard Form. 

IV.  LOGARITHMS 45 

Properties  of  Logarithms.  Use  of  the  Table  of 
Logarithms. 

V.  SMALL  MAGNITUDES 49 

Negligible  Magnitudes.    Properties  of  Small  Mag- 
nitudes.   Recapitulation  of  Formulae. 


x  CONTENTS 

VI.  THE  SLIDE  RULE 54 

Addition  with  Two  Scales.  Multiplication  with 
Logarithmic  Scales.  The  Construction  of  the 
Slide  Rule.  Checking  by  Approximation. 
Multiplication  and  Division.  Ratio  and  Pro- 
portion. Reciprocals.  Powers  and  Roots. 
Circular  Functions. 

VII.  GRAPHIC  REPRESENTATION 62 

Graphic  Diagrams.     Graphs  of  Equations. 

VIII.  GRAPHIC  ANALYSIS 68 

The  Graph  of  y  =  a  +  bx.  The  "Black-Thread" 
Method.  The  Graph  of  y  =  a  +  bx  +  cxz. 
Linear  Relationship  by  Change  of  Variables. 
Graphic  Interpolation.  Extrapolation. 

IX.  THE  PRINCIPLE  OF  COINCIDENCE 77 

Measurement  by  Estimation.  Measurement  by 
Coincidence.  The  Vernier.  Slide-Rule  Ratios. 

X.  MEASUREMENTS  AND  ERRORS 82 

Direct  and  Indirect  Measurements.  Independent, 
Dependent,  and  Conditioned  Measurements. 
Harmony  and  Disagreement  of  Repeated  Meas- 
urements. Errors  of  Measurements.  Classi- 
fication of  Errors.  Accidental  and  Constant 
Errors.  Errors  and  Variations.  Measurement 
of  Variates. 

XI.  STATISTICAL  METHODS 89 

Frequency  Distributions.  Class  Interval.  Types 
of  Frequency  Distribution.  The  Probability 
Curve.  Representative  Magnitudes.  The  Av- 
erage. The  Median.  The  Mode.  Choice  of 
Means.  Deviations.  Averages  by  Symmetry. 
Avera'ges  by  Partition.  Quartiles.  Semi- 
Interquartile  Range. 


CONTENTS  xi 

XII.  DEVIATION  AND  DISPERSION 100 

Characteristic  Deviations.  Total  Range.  Aver- 
age Deviation.  Standard  Deviation.  Disper- 
sion. Rule  for  the  Accuracy  of  the  Average. 
Slide-Rule  Determination.  Sigma  Notation. 
Dispersion  of  an  Average.  The  Statement  of 
a  Measurement.  Relative  Dispersion. 

XIII.  THE  WEIGHTING  OF  OBSERVATIONS 109 

Necessity  of  Weights  for  Observations.  Deter- 
mination of  Density  by  Different  Methods. 
Weights  for  Repeated  Values.  The  Weighted 
Average.  Arbitrary  Assignment  of  Weights. 
Weight  and  Dispersion.  Limitations  of  the 
Rule  w  =  k/dz.  Exception  to  the  Rule. 

XIV.  CRITERIA  OF  REJECTION     .      . 115 

Observational  Honesty.  Importance  of  Criteria. 
Chauvenet's  Criterion.  The  Probable  Error. 
Graphic  Approximation  to  Chauvenet's  Criter- 
ion. Irregularities  of  Small  Groups  of  Meas- 
urements. Justification  of  the  Criterion. 
Wright's  Criterion.  Comparison  of  Charac- 
teristic Deviations. 

XV.  LEAST  SQUARES  AND  VARIOUS  ERRORS  ....  123 
The  Average  as  a  Least-Square  Magnitude. 
Least  Squares  for  Conditioned  Measurements. 
Probable  Errors  of  Indirect  Measurements. 
Systematic  Errors.  Periodic  Errors.  Progres- 
sive Errors.  Constant  Errors. 

TABLES  139 


INTRODUCTION/ 


GENERAL  directions  and  advice  in  regard  to  study, 
observation,  experimentation,  care  of  apparatus,  char- 
acter and  arrangement  of  notes,  etc.,  will  be  given  at 
the  time  of  the  first  exercise  in  the  course  of  Laboratory 
Physics.  The  student  should  be  prepared  with  the  fol- 
lowing equipment: 

Material  Equipment.— The  note-book  should  be  of  the 
size  and  character  best  adapted  to  the  work.  It  can 
be  obtained  from  the  Department  of  Physics  or  else 
directions  will  be  given  as  to  what  kind  of  a  note-book 
should  be  used.  The  student  will  need  to  supply  him- 
self with  a  fountain  pen;  one  of  good  quality  will  last 
long  enough  to  make  its  cost  less  than  two  cents  per 
month.  A  piece  of  blotting  paper  should  be  obtained 
which  is  long  enough  to  reach  across  the  page  of  the 
note-book.  A  hard  pencil  with  the  point  kept  well  sharp- 
ened will  also  be  needed. 

Mental  Equipment. — The  student  should  have  a 
knowledge  of  algebra  as  far  as  the  solution  of  equations 
of  the  first  degree;  also  a  sufficient  knowledge  of  plane 
geometry  to  include  the  properties  of  perpendiculars, 
equal  triangles,  isosceles  and  similar  triangles,  the  area 
of  parallelograms  and  triangles,  the  theorem  of  Pytha- 
goras, and  the  properties  of  similar  figures.  An  intelli- 
gent comprehension  of  principles  is  as  important  as  a 
memory  of  rules  and  formulae.  He  should  not  have 


14  INTRODUCTION 

any  difficulty  in  applying  and  understanding  the  use 
of  letters  for  known  and  unknown  quantities,  symbols 
of  operation,  and  parentheses;  and  should  realize  that 
algebraical  identitieG  ure  true  for  whatever  numerical 
values  may  be  substituted.  He  should  be  able  to  solve 
an  equation  for  any  of  its  literal  components,  whether 
they  are  x,  y,  or  2,  or  any  other  letters  of  the  alphabet. 
He  should  have  a  knowledge  of  factoring,  the  reduc- 
tion of  fractions,  and  the  simplification  of  equations. 
The  fundamental  laws  of  exponents  should  be  known 
and  there  should  be  no  difficulty  in  dealing  with  nega- 
tive and  fractional  powers.  It  is  important  in  all 
physical  calculations  to  be  able  to  find  mentally  an 
approximate  value  for  a  numerical  formula,  and  there 
should  never  be  any  difficulty  in  pointing  off  the  pro- 
duct of  two  numbers  expressed  with  decimals,  without 
using  the  rule  for  the  number  of  decimal  places.  It  is 
particularly  important  that  the  student  should  appre- 
ciate the  facts  that  a  ratio  and  a  quotient  and  a  frac- 
tion are  all  the  same  thing,  that  a  proportion  is  an 
equation  in  which  one  fraction  is  equal  to  another,  that 
a  ratio  indicates  that  one  quantity  is  a  certain  number 
of  times  as  large  as  another,  that  this  "number  of  times" 
is  a  constant  for  both  sides  of  a  proportion  and  any 
proportion  can  be  written  in  the  form 


if 


that  x  and  y  must  be  proportional  if  x  always  equals 
c\y  for  any  values  of  x  and  y,  that  a  change  in  one  of 
two  quantities  which  are  proportional  must  be  accom- 


INTRODUCTION  15 

panied  by  an  equal  relative  change  in  the  other,  that 
two  quantities  which  are  inversely  proportional  have  a 
constant  product  instead  of  a  constant  quotient  and 
an  increase  in  one  is  necessarily  accompanied  by  an 
equal  relative  decrease  in  the  other,  etc.  A  knowledge 
of  the  rule  "  product  of  means  equals  product  of  ex- 
tremes" is  a  very  poor  substitute  for  an  understanding 
of  variation  and  proportion,  and  with  a  proper  com- 
prehension of  the  subject  there  is  no  need  of  even 
knowing  what  is  meant  by  such  expressions  as  "  pro- 
portion by  composition  and  division." 

The  student  should  read  each  of  the  following 
questions  and  answer  it  mentally  without  hesitation. 
If  it  is  necessary  to  "stop  and  think"  about  any  of 
them  he  should  make  a  note  of  the  ones  which  cause 
the  trouble  and  ask  the  advice  of  his  instructor  in  regard 
to  them.  This  will  often  make  a  great  difference  in 
the  ease  of  performing  the  later  laboratory  work. 

What  is  the  value  of  (m  —  x)  (m  +  x)? 
What  is  the  reciprocal  of  2/7? 
Reduce  0.395  to  a  percentage. 
What  per  cent,  is  .005? 
Write  the  cube  of  a  +  x. 
Solve  3  :  12  : :  16  :  x. 

Simplify  ||6- 

What  is  the  fourth  term  of  1000  :  100  :  :  31  :? 
Simplify  each  of  the  following: 

a7  x  5,  a7  + 5,  a7  X  a5,  a7  +  a5,  (a7)5,  (a5)7. 
Write  the  value  of  v. 
3  +  (4  -  4  -r-  2)  (7  X  4  -  3)  =? 

If  pv  is  a  constant  how  will  v  be  affected  by  doubling  p? 

4  4  x  7 

How  would  your  solve  3  +  .,  =  5  +  -"      — «-  ? 

'JC>  —  X    ~~"    O 


16  INTRODUCTION 

What  can  you  say  about   the  value  of  x  in    the   equation 


What  is  re2  -  7x  +  12  when  x  =  5? 

What  is  the  value  of  y  in  the  equation,  y  =  ax  +  b  when  x  =  0? 
Write  decimally,  and  add,  the  three  following  numbers  of 
tenths  of  an  inch:  9,  4,  17. 
Solve  3  :  2  :  :  x  :  100. 
Evaluate  (-  2)  (-  15)/(-  5). 
How  would  you  solve  1.31  x  -  .44  y  =  15.2? 
State  the  value  of  1/(1  +  re)  as  a  series. 
Solve  p  =  2ir\/l/g  for  I,  and  then  for  g. 
Which  is  the  larger,  37/147  or  38/148? 

In  the  equation  -  +  ^  =  1  what  is  the  value  of  either  unknown 
quantity,  x  or  y,  when  the  other  is  zero? 

Substitute  n  =  —  1  in  the  equation  (1  +  x)n  =  1  +  nx  +  .  .  . 

Substitute  n  =  x/2  in  the  same  equation  and  write  without 
fractional  exponents. 

Solve  42  :  x2  :  :  252  :  752. 

What  is  the  value  of  1  +  \  +  \  +  |  .  .  .  ? 

Solve  2  x  =  57°  17'  44". 

What  is  the  value  of  x*/x*.     Write  it  with  a  radical  sign. 

/  1  \3 
Reduce  (—  )    to  the  form  an. 

Multiply  .0011  by  .00011. 

Evaluate  ab/c  +  d  (e  +  /)  when  a  =  1,  b  =  2,  c  =  3,  d  =  4, 
e  =  5,  and/  =  6. 

Substitute  —  for  —  fo#  2/  m  l/(—  Jog  y)  and  simplify. 

?lf 

State  the  approximate  square  root  of  each  of  the  following  : 
2560,  256,  25.6,  2.56,  0.256. 

Simplify  or5'2,  x*+2,  x°~z,  41/2/4~2. 

Write  the  following  in  the  form  of  decimal  fractions:  |,   -,,   \, 

i,  I,  iV. 

What  do  you  know  about  the  right-hand  side  of  the  equation 
5  days  :  7  inches  :  :  ? 
1  *3  27  V  0  S1 
Does       3826/41       *  ^^  have  a  value  of  about  6,  or  about  (>(), 

or  about  600? 


INTRODUCTION 


17 


^  n  =  21   vsr 


does  n  change  if  I  becomes  j  of  its  former 

size?    If  t  becomes  j  as  large?    If  s  is  \  as  large? 
Fill  out  the  following  table: 


n 

n* 

2»-3 

1/n 

0 

1 

2 

3 

2x 

Physical  Arithmetic. — When  a  calculation  is  made 
with  numbers  obtained  from  physical  measurements 
the  final  result  never  needs  to  be  expressed  with  a 
greater  number  of  figures  than  the  original  data  con- 
tained. This  means  that  time  and  labor  can  be  saved 
by  abridging  the  customary  methods  of  "long"  multi- 
plication and  division  as  indicated  in  the  following- 
examples  : 


236453)6764309(28 . 60741 
472906 


230433)6764309(28.60741 
472906 


2035249 
1891624 

1436250 
1418718 

1753200 
1655171 


980290 
945812 

344780 
236453 

108327 


2035249 
1891624 

143625 
141872 

1753 
1655 

98 
95 

3 
2 


18  INTRODUCTION 

The  abridged  method  is  similar  to  the  ordinary  method 
except  that  the  divisor  and  dividend  are  made  to  fit 
each  other  not  by  stretching  out  the  dividend  with 
added  zeros  but  by  trimming  off  the  divisor.  The 
beginner  should  work  out  the  quotient  of  the  two 
numbers  given  above,  cancelling  the  right-hand  figure 
of  the  divisor  whenever  he  would  otherwise  "bring 
down"  a  zero,  but  not  referring  to  the  example  until 
he  has  finished.  After  the  second  figure  of  the  quotient 
has  been  obtained  the  problem  becomes,  not 

1436250  +  236453,  but  143625  +  236453. 

It  should  be  noticed  that  before  multiplying  23645 
by  6  a  quick  mental  determination  is  made  of  the 
amount  which  should  have  been  "carried"  from  the 
figure  that  was  last  cancelled.  6  X  3  =  18,  giving  1 
to  carry,  but  1 . 8  is  nearer  to  2  than  1  so  it  is  more 
accurate  to  carry  2  instead.  After  the  figure  4  of  the 
quotient  is  obtained  230433  is  to  be  multiplied  by  4. 
Ordinarily  it  is  sufficient  to  take  the  nearest  cancelled 
figure  and  say  4  X  6  =  24,  giving  2  to  carry;  but  as 
24  comes  close  to  the  number  25,  which  is  on  the 
boundary  between  2  to  carry  and  3  to  carry,  it  is 
well  to  investigate  one  more  cancelled  figure,  saying 
4  X  4  =  16;  about  2  to  carry;  4  X  6  -f-  2  =  26,  giv- 
ing 3  to  carry;  4  X  23  +  3  =  95. 


32.4761 
719.283 

32.4761 
719.283 

32U70Z 
719.283 

974283 
2598088 
649522 
2922849 
324761 
2273327 

2273327 
324761 
2922849 
649522 
2598088 
974283 

2273327 
32476 
29228 
650 
260 
10 

23359 . 5066363  23359 . 5066363  23359 . 5 1 


INTRODUCTION  19 

Three  methods  of  performing  a  multiplication  are 
given  above.  Examine  method  b  closely,  and  decide 
for  yourself  how  it  is  carried  out  and  whether  it  is 
perfectly  justifiable;  i.  e.,  whether  it  must  necessarily 
give  the  same  result  in  all  cases  as  method  a.  Repeat 
the  above  multiplication  as  in  method  b  except  that 
after  each  horizontal  line  of  partial  products  is  ob- 
tained the  right-hand  figure  of  the  multiplicand  is  to 
be  cancelled  and  the  next  partial  product  started 
directly  under  the  previous  ones.  Compare  the  result 
with  method  c. 

Make  up  and  practice  other  examples  of  abridged 
multiplication  and  division,  some  having  more  figures 
in  one  number  than  in  the  other.  It  is  very  important 
that  the  method  shall  be  clearly  understood,  as  the 
unabridged  methods  will  not  be  permissible  in  any  of 
the  work  done  in  the  laboratory  course. 


I.  WEIGHTS  AND  MEASURES 

Apparatus. — Scale  of  centimetres  and  millimetres; 
graduated  cylinder;  graduated  pipette;  irregular 
solid;  platform  balance;  set  of  weights  from  1  gm.  to 
500  gm.;  set  of  avoirdupois  weights,  1  ounce  to  8 
ounces;  small  test-tube. 

C.  G.  S.  System. — In  all  scientific  work  the  older 
weights  and  measures,  such  as  the  cubit,  palm,  pace,  or 
foot,  have  been  superseded  by  a  single  system  of 
weights  and  measures  which  is  now  in  universal  use  in 
all  civilized  countries.  It  is  usually  spoken  of  as  the 
C.  G.  S.  system,  from  the  initial  letters  of  the  units 
of  length  (the  centimetre),  of  mass  (the  gram),  and  of 
time  (the  second).  These  three  units  are  called  " funda- 
mental," and  all  of  the  other  units  of  the  system  have 
been  so  chosen  as  to  depend  upon  these  three  in  as 
simple  a  manner  as  possible.  For  example,  the 
" derived"  unit  of  velocity  is  such  as  will  denote  move- 
ment through  a  single  unit  of  distance  in  a  single  unit 
of  time;  the  density  which  is  taken  as  numerically 
equal  to  one  is  the  density  of  such  a  substance  as  will 
weigh  one  gram  for  each  cubic  centimetre  of  its  volume ; 
the  unit  of  force  is  the  force  that  must  act  for  a  unit  of 
time  in  order  to  produce  a  change  of  one  unit  in  the 
velocity  of  a  unit  mass. 

Unit  of  Length. — The  scientific  unit  of  length  is  the 
centimetre.  It  is  equal  to  about  half  a  finger-breadth 
and  is  often  to  be  found  on  tape-measures,  rulers,  etc. 


WEIGHTS  AND  MEASURES  21 

These  are  simply  copies  of  accurate  standards  belong- 
ing to  the  manufacturers,  which  in  turn  owe  their 
accuracy  to  a  careful  comparison  with  the  standards  of 
the  government.  In  the  case  of  the  governments  that 
subscribed  to  the  Metric  Convention,  including  the 
United  States,  the  standards,  called  national  prototype 
metres,  are  lengths  of  one  metre  (100  centimetres) 
carefully  laid  off  between  lines  near  the  ends  of  cer- 
tain bars  of  platinum-iridium  alloy  which  are  rather 
more  than  a  metre  long  and  have  a  cross-section  that 
somewhat  resembles  a  letter  X.  The  greater  length  is 
used  instead  of  a  single  centimetre  because  it  can  be 
measured  more  accurately.  The  standards  were  con- 
structed at  Paris  and  distributed  by  lot  among  the 
signatories  to  the  Metric  Convention  about  1889, 
after  being  carefully  compared  with  one  another  so 
that  their  relative  errors  were  accurately  known.  One 
of  these,  which  is  kept  at  the  International  Bureau 
of  Weights  and  Measures,  near  Paris,  is  known  as 
the  international  prototype  metre  and  gives  the  same 
length  as  the  original  flat  platinum  bar  constructed  for 
the  French  Government  by  Borda  and  called  the 
metre  des  archives.  The  Borda  standard  was  intended 
to  equal  one  ten-millionth  of  the  length  of  the  meridian 
quadrant  passing  through  Paris  from  the  north  pole 
to  the  equator,  the  earth  itself  thus  furnishing  the 
original  standard.  The  metal  bar,  however,  is  now 
taken  as  the  fundamental  standard,  not  only  because 
a  microscopic  measurement  of  it  can  be  performed 
more  easily  and  more  accurately  than  a  geodetic  sur- 
vey, but  also  because  the  actual  length  of  the  earth's 
quadrant  is  continually  changing.  Its  average  length, 


22  WEIGHTS  AND  MEASURES 

according  to  the  best  estimations,  is  about  10,002,100 
metres. 

The  multiples  and  subdivisions  of  the  metre  that 
are  in  actual  use  are  the  kilometre  (1  km.  is  1000 
metres,  or  about  f  of  a  mile),  the  centimetre  (1  cm),  the 
millimetre  (0.1  cm),  and  the  micron  (0.0001  cm). 

Units  of  Area  and  Volume. — The  scientific  unit  of 
area  is  the  square  centimetre  (1  cm2),  the  area  of  a 
square  each  of  whose  sides  is  1  cm.  in  length.  A 
square  foot  is  about  1000  cm2.  The  unit  of  volume 
is  the  cubic  centimetre  (1  cm3),  the  volume  of  a  cube 
that  measures  one  centimetre  on  each  edge.  The  dry 
and  liquid  quarts  are  both  in  the  neighborhood  of 
1000  cm3. 

Units  of  Mass  and  Density. — The  scientific  unit  of 
mass,  or  for  practical  purposes  the  unit  of  weight  in 
vacuo  at  sea  level,  lat.  45°,  is  the  gram  (1  gm.).  It 
is  derived  from  kilogram  prototype  standards  (1000 
gm.  =  1  kgm.  =  2.2  Ibs.)  established  at  the  same  time 
as  the  standards  of  length  and  was  originally  intended 
to  be  the  mass  of  one  cubic  centimetre  of  water  under 
standard  conditions.  More  careful  measurements,  how- 
ever, on  water  free  from  dissolved  air  have  shown  that 
even  at  the  temperature  of  its  greatest  density  (3.98° 
C.)  a  gram  of  water  occupies  a  trifle  more  space  than 
one  cubic  centimetre,  although  the  excess  is  only  one- 
seventieth  as  great  as  it  is  at  room  temperature.  In 
cases  where  the  expansion  of  water  can  be  neglected 
it  may  be  considered  that  its  density,  or  ratio  of  mass 
to  volume,  is  unity. 

Unit  of  Time. — The  scientific  unit  of  time  is  the 
second,  the  1/86400  part  of  the  length  of  an  average 


WEIGHTS  AND  MEASURES  23 

day  from  noon  to  noon.  As  the  length  of  the  solar 
day  varies  at  different  seasons  of  the  year  the  second 
is  practically  determined  as  1/86164.1  of  the  time  of 
a  complete  rotation  of  the  earth  with  respect  to  the 
fixed  stars.  Fairly  accurate  seconds  can  be  laid  off 
by  counting,  at  an  ordinary  conversational  speed, 
"one  thousand  and  one,  one  thousand  and  two,  one 
thousand  and  three,"  etc. 

Practice  in  Using  the  C.  G.  S.  System. — Across  the 
top  of  one  page  of  the  note-book  draw  a  horizontal 
line  just  ten  centimetres  long  and  rule  two  short  per- 
pendicular lines  across  its  ends  to  establish  the  length 
more  definitely.  Along  the  right-hand  edge  of  the 
page,  draw  a  line  twenty-five  centimetres  long.  Under 
the  first  line  draw  five  others  of  various  lengths  with- 
out measuring  them.  After  they  have  been  drawn 
measure  each  one  with  a  scale  of  centimetres  and  milli- 
metres, and  record  its  length  to  the  nearest  millimetre. 
For  example,  if  the  length  appears  to  be  about  172J 
mm.  it  should  be  recorded  as  17.2  cm.;  if  about  172f 
mm.  it  should  be  called  17.3  cm. 

Rule  for  "Rounding  Off"  One-Half.— If  it  is  impos- 
sible to  decide  between  17.2  and  17.3  the  preferred 
rule  is  to  record  the  nearest  even  number  rather  than 
the  odd  number  that  is  equally  near.  In  a  series  of 
several  measurements  this  procedure  will  be  as  apt  to 
riiake  a  record  too  large  as  to  make  one  too  small,  and 
the  average  of  several  such  values  will  have  only  a 
very  slight  error,  if  any.  If  the  rule  were  that  the 
half  should  be  regularly  increased  to  the  next  larger 
unit  the  errors  would  not  balance  one  another  and  the 
average  would  tend  to  be  brought  up  to  a  larger  value 


24  WEIGHTS  AND  MEASURES 

than  it  should  have.  The  same  advantage  could  of 
course  be  obtained  by  always  using  the  odd  number, 
but  the  even  number  has  one  slight  additional  merit, 
namely,  that  in  case  it  should  have  to  be  divided  by 
two  a  recurrence  of  the  same  situation  would  be  avoided. 

By  comparison  with  the  lines  already  drawn  make 
a  mental  estimate  of  the  length  and  width  of  the  note- 
book; then  verify  the  estimate  by  measuring  with  a 
scale.  Remember  to  record  clearly  all  experimental 
work  that  is  done;  thus,  the  completed  notes  should 
show  at  a  glance  which  number  is  the  actual  width 
and  which  is  the  rough  estimate. 

The  Hand  as  a  Measure.— Hold  your  hand  across  a 
centimetre  scale  and  either  spread  the  fingers  slightly 
or  crowd  them  closer  together,  as  may  be  necessary, 
so  as  to  make  a  whole  number  of  finger-breadths 
equal  to  a  whole  number  of  centimetres.  Then  hold 
the  hand  in  a  similar  position  while  using  it  for  practic- 
ing approximate  measurements  of  various  objects. 
Record  also  the  exact  measurements  of  the  same  ob- 
jects as  they  are  obtained  afterward  with  the  grad- 
uated scale. 

Separate  the  thumb  and  little  finger  as  far  as  can 
be  conveniently  done  without  special  effort  and  measure 
your  span  in  centimetres  and  millimetres.  Repeat 
this  measurement  five  times,  being  careful  not  to  let 
the  sight  of  the  scale  under  your  hand  influence  the 
extent  to  which  the  fingers  are  spread,  and  decide 
which  is  the  most  satisfactory  value.  Measure  the 
breadth  of  the  table  by  means  of  successive  spans, 
using  finger-breadths  for  the  final  fraction  of  a  span, 
and  compare  the  result  with  the  actual  breadth. 


WEIGHTS  AND  MEASURES  25 

Measurement  of  Area. — Ask  the  instructor  to  draw 
an  irregular  outline  in  your  note-book.  Count  the  num- 
ber of  squares  of  the  ruled  paper  which  are  entirely 
included  within  it,  and  to  this  add  half  the  number 
of  the  squares  that  are  cut  by  the  boundary  of  the 
figure.  The  result  will  be  the  area  of  the  irregular 
figure,  not  in  square  centimetres,  of  course,  but  in 
terms  of  the  small  ruled  squares.  Try  to  obtain  a 
more  accurate  value  for  the  total  area  by  estimating 
as  closely  as  possible  how  many  tenths  of  each  small 
square  which  is  cut  through  by  the  curve  are  included 
within  it.  Why  does  the  first  result  agree  so  closely 
with  the  second? 

On  the  same  irregular  figure  block  off  an  equal  area 
by  drawing  several  rectangles  and  triangles  to  cover 
it.  If  a  corner  of  a  triangle  projects  considerably 
beyond  the  irregular  line  draw  one  side  of  the  triangle 
so  that  it  includes  less  than  the  requisite  amount  and 
try  to  make  the  two  opposite  errors  balance  as  nearly 
as  possible.  Then  find  the  total  area  of  the  geometrical 
figures,  measuring  their  dimensions  not  by  means  of 
the  centimetre  scale  but  with  a  scale  copied  from  the 
ruling  of  the  note-book  or  by  transferring  each  length 
to  the  ruled  page  with  a  pair  of  dividers,  so  as  to  obtain 
the  result  in  the  same  units  as  before. 

Measurement  of  Volume. — Examine  the  graduated 
cylinder  and  pipette;  compare  the  indicated  volumes 
with  an  imaginary  cube  built  up  on  one  centimetre  of 
the  graduated  scale.  Pour  into  a  test-tube  an  amount 
of  water  which  you  consider  to  be  10  cm3;  then  measure 
it»carefully. 

Put  an  irregular  block  of  metal  into  a  graduated 


26  WEIGHTS  AND  MEASURES 

cylinder  partly  filled  with  water  and  determine  its 
volume  by  the  change  in  the  water  level.  In  reading 
a  cylinder  or  a  pipette  the  most  accurate  result  is 
obtained  by  noting  the  height  of  the  lowest  part  of  the 
surface  of  the  liquid. 

Measurement  of  Mass. — Examine  the  brass  weights 
in  a  set  extending  from  1  gram  to  500  grams,  and 
observe  especially  the  size  of  the  10-gram  weight. 
What  would  its  volume  be  if  its  density  were  ten? 
Is  its  volume  greater  or  less  than  this  if  the  density 
of  brass  is  8.5? 

Examine  the  platform  balance  and  notice  the  two 
wooden  wedges  that  raise  the  pans  so  as  to  keep  their 
weight  off  from  the  accurately  ground  bearings  when 
the  apparatus  is  not  in  use.  Remove  the  wedges 
carefully  and  notice  where  the  moving  pointer  finally 
comes  to  rest  on  the  scale.  Notice  the  counterpoise, 
which  can  be  screwed  toward  one  side  or  the  other  in 
order  to  adjust  the  position  of  equilibrium,  but  do 
not  attempt  to  move  it  unless  you  are  sure  that  the 
apparatus  is  level  and  the  scale  pans  are  clean.  Note 
the  sliding  weight  that  is  used  for  weighing  fractions 
of  a  gram. 

Find  the  mass,  in  grams,  of  a  four-ounce  avoirdupois 
weight,  and  determine  for  yourself  why  it  is  advisable 
to  put  the  object  to  be  weighed  on  the  left-hand  scale 
pan  rather  than  on  the  right-hand  one. 

Measurement  of  Density. — Find  the  volume  of  the 
200-gram  weight  by  measuring  its  height  and  diameter 
as  carefully  as  possible,  but  do  not  immerse  it  in  water. 
Imagine  the  handle  of  the  weight  to  be  flattened  out 
and  spread  over  the  top  of  the  cylindrical  part  and 


ANGLES  AND  CIRCULAR  FUNCTIONS  27 

estimate  as  well  as  you  can  how  much  this  would  add 
to  the  height  of  the  cylinder.  Then  calculate  the 
density  of  the  weight,  using  the  abridged  methods  of 
multiplication  and  division.  The  result  may  turn  out 
to  be  less  than  the  usual  density  of  brass  if  the  weight 
consists  of  two  parts  with  an  air-space  between  them. 
Find  the  mass  also  of  the  irregular  solid  whose  vol- 
ume was  determined  by  immersion,  and  determine  its 
density. 

II.  ANGLES  AND  CIRCULAR  FUNCTIONS 

Apparatus. — A  pair  of  dividers;  a  protractor  pro- 
vided with  a  "  diagonal  scale." 

Unit  of  Angle. — The  practical  unit  of  angle  is  the 
degree,  and  a  right  angle  contains  90  of  these  units. 
Each  degree  is  subdivided  into  sixty  minutes,  and 
each  of  these  is  in  turn  made  up  of  60  still  smaller 
units,  or  seconds.  In  scientific  usage,  however,  an 
angle  of  a  given  size  is  considered,  not  as  a  number  of 
units,  but  as  the  ratio  of  one  such  number  of  units, 
which  represents  a  certain  length,  to  another  number 
of  units,  which  represents  a  different  length.  This 
means  that  it  is  not  expressed  in  any  arbitrary  concrete 
unit  whatever,  but  as  an  abstract  number.  The  reason 
for  this  can  be  understood  if  a  length  of  1  centimetre 
and  an  angle  of  90  degrees  are  drawn  on  a  sheet  of  paper 
and  observed  through  a  niagnifying  glass.  The  centi- 
metre may  now  appear  to  be  two  centimetres  in  length, 
but  the  angle  of  90  degrees  does  not  become  180 
degrees,  it  remains  exactly  as  large  as  before.  The 
same  thing  is  of  course  true  of  a  pure  number:  with 


28  ANGLES  AND  CIRCULAR  FUNCTIONS 

a  very  slight  magnification  two  objects  may  both  be 
made  to  look  larger,  but  no  amount  of  magnifying 
power  will  make  them  look  like  three. 

Numerical  Measure  of  an  Angle.— If  a  circle  is 
drawn  with  the  vertex  of  any  angle  as  its  centre  the 
arc  intercepted  by  the  angle  will  have  a  certain  length 
as  compared  with  the  radius,  and  this  ratio  of  arc  to 
radius  will  be  the  same  whether  the  circle  is  large  or 
small.  It  will  depend  only  upon  the  size  of  the  angle, 
and  so  can  be  used  as  a  measure  of  it.  The  angle, 
then,  which  is  equal  to  the  number  one  must  be  such 
an  angle  that  the  corresponding  arc  and  radius  are 
of  the  same  length.  It  is  also  clear  that  an  angle 
which  extends  entirely  around  a  point,  that  is,  four 
right  angles  or  360  degrees,  must  have  2  TT  for  the 
ratio  of  its  arc,  the  circumference,  to  the  radius;  and 
if  2  TT  is  equal  to  360°  the  angle  TT  must  be  180°,  and 
the  angle  1  will  be  180%r. 

Practice  translating  such  numbers  as  the  following 
into  degrees  until  it  can  be  done  without  any  hesitation : 

i  7T    =    ?         27T    =    ?         i  7T    =    ?        4  7T    =    ?        7T/3     =    ?         f  T    «    ? 

7T/4    =    ?        X/7T    =    ? 

Work  out  the  number  of  degrees  in  the  unit  angle  by 
the  abridged  method  of  division.  Is  the  angle  3  greater 
or  less  than  180  degrees? 

Construction  and  Measurement  with  the  Protractor. 
—Examine  the  protractor  and  the  scale  of  degrees 
that  is  engraved  on  it.  Notice  the  line  joining  0° 
and  180°,  and  decide  exactly  where  the  vertex  of  the 
indicated  angles  must  be  located  on  the  instrument. 

Draw  a  horizontal  line  in  the  note-book  and  mark 


ANGLES  AND  CIRCULAR  FUNCTIONS  29 

any  point  on  it  with  a  short  cross-line.  How  can  you 
use  the  protractor  to  draw  a  line  which  passes  through 
this  point  and  makes  a  given  angle  with  the  horizontal 
line? 

Draw  a  triangle  whose  angles  are  ir/2,  ir/3,  ir/Q. 
Draw  another  so  that  its  angles  are  ?r/4,  7r/4,  7r/2. 

The  Diagonal  Scale. — If  the  protractor  is  provided 
with  what  is  known  as  a  diagonal  scale  notice  that 
at  the  bottom  of  this  scale  there  is  a  horizontal  line 
which  is  divided  into  centimetres  or  inches,  and  that 
one  division  at  the  end  of  the  scale  is  subdivided 
into  millimetres  or  tenths  of  an  inch.  Notice  that 
any  number  of  centimetres  and  tenths,  within  limits 
that  depend  upon  the  total  length  of  the  scale,  can  be 
found  already  laid  off  as  a  single,  continuous  stretch 
of  the  base  line,  the  millimetres  being  measured  from 
the  proper  point  to  the  junction  of  the  millimetre 
scale  and  the  centimetre  scale,  and  the  centimetres 
then  extending  onward  the  required  distance  beyond 
the  junction  point. 

Apply  a  pair  of  fine  dividers  to  the  ends  of  some  ob- 
ject that  is  not  longer  than  the  diagonal  scale;  then 
transfer  them  to  the  scale  and  measure  their  spread 
to  the  nearest  millimetre.  They  should  always  be 
held  flat  to  avoid  marring  the  scale. 

Notice  that  there  are  ten  other  lines,  spaced  at 
equal  distances  and  parallel  to  the  base  line,  and  that 
they  are  intersected  by  certain  diagonal  lines.  Observe 
one  of  these  that  forms  the  hypothenuse  of  a  large 
right-angled  triangle  and  determine  the  length  of  its 
shortest  side.  If  the  parallel  lines  are  equidistant 
what  are  the  lengths  of  their  short  segments  that  are 


30  ANGLES  AND  CIRCULAR  FUNCTIONS 

included  in  the  large  triangle  and  cut  it  up  into 
smaller  similar  triangles?  Determine  for  yourself  how 
it  is  that  this  arrangement  will  show  any  number  of 
centimetres  and  tenths  and  hundredths  of  a  centi- 
metre as  a  single  distance  on  one  of  the  horizontal  lines, 
laid  off  between  two  intersecting  lines.  Practice  using 
the  dividers  and  diagonal  scale,  both  for  laying  off 
predetermined  lengths  and  for  measuring  unknown 
distances,  until  you  are  perfectly  familiar  with  the 
method  of  procedure. 

An  angle  may  be  identified  or  measured  in  other 
ways  than  by  the  number  of  degrees  that  it  contains 
or  by  the  ratio  of  arc  to  radius.  In  case  one  side  of  the 
angle  is  horizontal  a  given  slope  or  inclination  of  the 
other  side  will  always  correspond  to  a  definite  angle. 
The  usual  method  of  indicating  a  slope  is  by  means  of 
the  ratio  of  height  to  base-line,  or  amount  of  vertical 
distance  for  a  unit  amount  of  horizontal  distance. 

The  Tangent  of  an  Angle. — Thus,  in  the  figure,  the 
slope  OA  is  numerically  given  by  the  ratio  AB  :OB, 

or  CD  -T-  OD,orEF/OF 


A 


-all    obviously    equal 


to  one  another  and  to 
about  0.39 — and  it  is 
said  that  OA  has  a  39 
per  cent,  slope,  or  that 

the  tangent  of  Z_  AOB  is  0.39.  The  reason  for  calling 
this  quantity  the  "  tangent "  of  the  angle  may  be  under- 
stood by  noticing  that  the  line  drawn  between  the  two 
sides  of  the  angle  and  tangent  to  the  arc  at  one  end  will 
be  numerically  equal  to  the  ratio  in  question  if  the  radius, 
OD  or  OE,  is  of  unit  length,  for  then  CD /OD  =  CD /I  = 


ANGLES  AND  CIRCULAR  FUNCTIONS  31 

CD.  If  the  radius  is  not  of  unit  length  the  tangent  line 
will  of  course  be  greater  or  less  than  the  numerical 
tangent  of  the  angle.  In  general,  if  A  is  an  acute 
angle  of  a  right-angled  triangle  the  ratio  of  the  side 
opposite  this  angle  to  the  adjacent  side  is  called  the 
tangent  of  the  angle  A,  and  is  customarily  abbreviated 
to  tan  A. 

With  the  protractor  draw  carefully  an  angle  of  58 
degrees,  and  a  perpendicular  to  one  of  its  sides.  Meas- 
ure the  tangent  and  see  how  close  your  experimental 
result  comes  to  the  theoretical  value  1 . 600.  What  is 
the  numerical  value  of  tan  45°?  tan  0°?  tan  7r/4? 
Draw  angles  of  80°,  85°,  88°,  89°,  remembering  that 
the  base  line  can  always  be  made  short  enough  to 
bring  the  perpendicular  within  the  limits  of  the  paper; 
then  decide  what  the  numerical  value  must  be  for 
tan  90°.  How  could  you  find  the  tangent  of  the  angle 
OAB  on  page  30?  What  is  its  approximate  value? 
What  relation  is  there  between  the  numerical  values 
of  tan  OAB  and  tan  AOB? 

Table  of  Tangents. — Near  the  bottom  of  the  next 
unused  page  of  your  note-book  draw  a  horizontal 
line  of  20,  25,  or  50  squares  in  length,  following  one 
of  the  blue  ruled  lines  and  extending  nearly  across  the 
page.  Call  its  length  unity  (1),  erect  a  perpendicular 
at  its  right-hand  end,  and  on  the  perpendicular  lay 
off  heights  equal  to  tan  10°,  tan  20°,  tan  30°,.  .  .,  as 
far  as  the  length  of  the  page  will  permit,  using  the 
values  obtained  from  the  table  on  page  141.  Opposite 
the  number  of  degrees  in  the  column  headed  DEG  will 
be'found  the  corresponding  tangent,  with  the  decimal 
point  omitted,  in  the  column  TAN.  From  45°  to  90° 


32  ANGLES  AND  CIRCULAR  FUNCTIONS 

the  numbers  run  upward  in  the  right-hand  columns 
over  DEG  and  TAN  at  the  bottom  of  the  page.  If  there 
is  any  trouble  in  understanding  the  arrangement  of  the 
table  use  it  to  verify  the  following  equations  before 
proceeding  further: 

Tan  1°  =  .0175;  tan  2°  =  .0349;  tan  6°  =  .1051;  tan  44°  = 
.9657;  tan  45°  =  1.000;  tan  46°  =  1.036;  tan  84°  =  9.514;  tan 
85°  =  11.43;  tan  89°  =  57.29. 

Draw  slant  lines  from  the  points  marked  off  on  the 
vertical  line  to  the  left-hand  end  of  the  base  line  and 
test  the  angles  with  the  protractor  to  see  that  they  are 
10°,  20°,  ....  If  mistakes  have  been  made  repeat 
the  construction  on  the  next  page;  do  not  correct 
the  first  diagram  by  erasures. 

The  Sine  of  an  Angle. — With  the  base  line  as  a  radius 
draw  an  arc  about  its  left-hand  end  extending  from 
0°  to  90°.  Complete  the  series  of  angles  to  90°  by 
laying  off  ten-degree  arcs  with  a  pair  of  dividers. 
Find  the  point  where  the  arc  intersects  the  line  whose 
slope  is  10  degrees,  and  measure  carefully  the  verti- 
cal distance  from  this  point  down  to  the  base  line.  If 
the  diagram  has  been  carefully  drawn  the  value  should 
come  out  0.174;  remembering  always  that  the  unit 
of  measurement  is  not  the  small  square  but  the  whole 
base  line.  This  number  is  less  than  tan  10°,  and  cor- 
responds to  EF/OD  in  the  previous  diagram,  or  to  its 
equivalent,  EF/OE.  This  ratio  of  height  to  slant 
distance  is  another  quantity  which  is  perfectly  definite 
for  a  given  angle  and  is  called  the  "sine"  of  the  angle. 
For  any  angle  not  greater  than  90°  we  may  define 
11  sin  A,"  as  it  is  usually  abbreviated,  as  the  ratio  of 


ANGLES  AND  CIRCULAR  FUNCTIONS  33 

the  opposite  side  to  the  hypothenuse,  in  a  right- 
angled  triangle  similar  to  the  one  used  for  denning 
tan  A. 

In  the  same  way  measure  sin  20°,  sin  30°,  .  .  .  sin 
90°,  or  as  many  of  them  as  may  be  directed  by  the 
instructor;  and  tabulate  the  results  as  shown  in  the 
margin,  where  sin  20°  is  represented  as  having  been 
measured  and  found  to  be  .344,  and  then  corrected  by 
comparison  with  the  table  on  page 


141,  which  gives  it  as  .344—  .002,      angle 
or  .342.     Notice  that  the  sine,  like 
the  tangent,  is  to  be  read  upward        10' 
from  the  bottom  of  the  page  for 
angles  greater  than  45°.     After  your        40 


0 


20' 
30 ' 


sine 


0 

,174±     0 
344 -.002 
500±     0 
640 +.003 


table  of  sines  has  been  completed 
find  the  angle  whose  sine  has  the  largest  correction, 
and  divide  this  correction  by  the  true  value  of  the 
sine  in  order  to  find  the  relative  error  of  your  measure- 
ment. Thus,  if  the  quotient  is  .0024  your  measurement 
has  an  error  of  24  parts  out  of  10,000,  or  about  J  of  1%. 

Definition  of  Function. — The  sine  and  the  tangent  of 
an  angle  are  called  " functions"  of  the  angle,  the 
expression  function  of  a  variable  magnitude  meaning 
simply  a  second  quantity  which  has  a  definite  value 
for  each  value  of  the  first  magnitude.  For  example, 
x2  —  3x  +  2  is  called  a  'function  of  x,  because  it  has 
a  definite  value  for  any  definite  value  which  may  be 
assigned  to  x.  Similarly  V  x,  logarithm  of  x,  ax,  tan  x, 
are  all  functions  of  x}  as  is  also  any  other  algebraical 
formula  which  involves  x. 

The  Cosine  of  an  Angle. — The  only  other  function  of 
an  angle,   or  "circular"  function,"   in  frequent  use  is 
3 


34         ACCURACY  AND  SIGNIFICANT  FIGURES 

the  cosine,  corresponding  to  OF,  or  more  strictly  to 
OF/OE,  in  the  figure  above.  It  is  the  ratio  of  adjacent 
side  to  hypothenuse,  and  in  the  note-book  diagram  is 
the  left-hand  segment  of  the  base  line  which  is  cut  off 
by  the  vertical  sine-line. 

From  the  diagram  in  your  note-book  find  the  angle 
whose  cosine  is  \  by  locating  the  point  on  the  arc  that 
is  directly  above  the  centre  of  the  base  line.  What  is 
cos  0°?  cos  90°?  Verify  your  statements  by  reference 
to  the  table  on  page  141. 


III.    ACCURACY  AND  SIGNIFICANT  FIGURES 

Apparatus. — Scale  of  centimetres  and  millimetres; 
card  or  strip  of  paper;  circular  brass  disc  for  measuring 
a  circumference. 

Estimation  of  Tenths. — In  the  physical  laboratory 
all  measurements  are  to  be  made  as  accurately  as 
possible  unless  there  is  some  evident  reason  for  mak- 
ing them  more  roughly.  Measurements  with  the  metre 
stick  should,  of  course,  be  expressed  to  the  nearest 
millimetre  rather  than  the  nearest  centimetre.  This 
is  not  the  limit  of  accuracy,  however.  With  a  little 
care  it  is  not  difficult  to  imagine  each  millimetre 
divided  into  ten  equal  parts  and  to  estimate  just  how 
many  of  these  parts  are  included  in  the  length  that  is 
to  be  measured.  Experienced  observers  even  attempt 
to  determine  hundredths  of  a  millimetre  in  the  same 
way,  and  find  that  one  man's  estimate  will  hardly 
ever  differ  from  another's  by  more  than  one  or  two 
hundredths.  The  estimation  of  hundredths  of  the 
smallest  scale  divisions,  however,  is  chiefly  of  import- 


ACCURACY  AND  SIGNIFICANT  FIGURES        35 

ance  to  the  expert;  the  beginner  will  find  that  even 
the  estimation  of  tenths  is  a  rather  uncertain  process, 
but  proficiency  can  be  gained  rapidly  by  experiment- 
ing with  the  larger  subdivisions  of  a  scale,  where  the 
determinations  can  afterward  be  verified. 

Draw  a  short  line  at  right  angles  to  the  edge  of  a 
card  or  slip  of  paper  and  hold  this  edge  on  a  scale  of 
centimetres  and  millimetres  in  such  a  way  that  the 
smallest  graduations  are  hidden  but  the  marks  indi- 
cating centimetres  and 
half-centimetres  are  vis- 
ible.  Mentally  divide 
the  half-centimetre  in 
which  the  cross-line 

comes  into  five  equal  parts  by  four  imaginary  lines, 
and  in  this  way  estimate  the  scale  reading  to  the 
nearest  millimetre.  In  the  figure  the  line  seems  to 
be  either  three  fifths  or  four  fifths  of  the  way  from  the 
scale-division  19.5  to  20.0,  but  it  may  be  difficult  to 
decide  which  is  the  nearer  without  actual  measure- 
ment. In  practice,  however,  the  estimate  should  be 
written  down  and  the  card  then  be  allowed  to  slide 
carefully  across  the  ruler  until  the  millimetre  scale  is 
just  exposed.  The  estimate  is  afterward  to  be  verified 
or  corrected,  as  the  case  may  require. 

Make  ten  such  estimates  with  the  card  placed  any- 
where along  the  scale  at  random,  tabulating  the 
determinations  and  the  verifications  in  parallel  columns. 
Then  hold  the  card  a  trifle  higher,  so  as  to  hide  the 
half -centimetres  as  well  as  the  millimetres  and  make 
twenty  more  determinations;  these  will  also  be  esti- 
mates of  the  nearest  millimetre,  but  will  require  the 


36        ACCURACY  AND  SIGNIFICANT  FIGURES 

more  difficult  process  of  deciding  upon  tenths  of  a  whole 
centimetre  instead  of  fifths  of  a  half-centimetre. 

Mistakes  in  Estimating  Tenths.— Omitting  the  pre- 
liminary estimate  of  fifths,  examine  the  table  of  esti- 
mated tenths  closely  and  find  out  what  kind  of  error 
you  are  most  apt  to  make.  Some  students  find  it 
hardest  to  estimate  0.3  and  0.7  correctly;  others 
have  almost  a  uniform  tendency  to  read  a  position 
like  12.0  as  either  11.9  or  12.1.  The  latter  mistake 
is  due  to  the  fact  that  a  minute  deviation  from  the 
position  of  a  visible  graduation  is  very  easily  noticed 
and  there  is  a  tendency  to  consider  it  as  a  single  tenth. 
Of  course  if  it  amounts  to  more  than  half  of  a  tenth 
this  is  correct;  but  if  it  is  less  than  half  a  tenth  it 
should  be  considered  as  0.0  instead  of  0.1.  The  same 
bias  may  even  cause  a  tendency  to  read  0.1  and  0.9 
as  0.2  and  0.8.  On  the  other  hand  there  may  be  just 
the  opposite  error  if  the  graduated  lines  are  rough  or 
coarse,  unless  the  student  is  careful  to  estimate  from 
the  imaginary  centre  of  such  a  "line"  instead  of  from 
its  margin. 

If  a  definite  kind  of  error  is  evident  from  a  study  of 
your  table  see  if  it  can  be  overcome  when  making 
another  short  series  of  determinations.  Then  draw  a 
second  line  on  the  card,  place  the  latter  in  position  as 
before,  estimate  both  points,  and  find  the  distance 
between  them  by  subtraction.  This  is  the  customary 
method  of  measuring  a  length,  and  is  preferable  to 
making  one  line  coincide  exactly  with  a  scale  division 
and  estimating  only  the  other  one,  in  spite  of  an 
obvious  additional  source  of  error. 


ACCURACY  AND  SIGNIFICANT  FIGURES        37 

Value  of  TT. — Make  an  experimental  determination 
of  the  value  of  the  constant,  TT,  by  rolling  the  brass 
disc  along  the  graduated  surface  of  a  metre  stick  in 
order  to  determine  its  circumference  by  noticing  the 
readings,  to  a  tenth  of  millimetre,  where  the  marked 
radius  touches  the  scale  at  the  beginning  and  the  end 
of  the  measurement.  Measure  the  diameter  with 
especial  care,  noting  down  centimetres,  tenths,  and 
hundredths;  divide  the  circumference  by  it,  and  see 
how  many  figures  of  your  result  agree  with  the  theoret- 
ical 3.141592653589793238462643383279502884197169. 

Physical  Measurement. — The  operation  of  making  a 
measurement  is  merely  counting;  it  is  the  determina- 
tion of  how  many  units  are  required  in  order  to  be 
equal  to  a  given  quantity  of  the  same  kind.  But  while 
a  count  such  as  a  census  of  the  number  of  individuals 
in  a  town  must  give  a  perfectly  definite  whole  number 
it  usually  happens  that  a  physical  measurement  will 
not  give  a  whole  number,  or  even  a  commensurable 
number  except  as  the  result  of  an  error,  and  successive 
repetitions  of  a  measurement  will  give  a  number  of 
different  apparent  values.  Accordingly,  any  numerical 
statement  of  a  measurement  must  be  merely  an  approxi- 
mation to  an  unknown  truth,  and  will  be  perceptibly 
correct  or  perceptibly  incorrect  according  to  how 
closely  it  is  examined. 

Ideal  Accuracy. — A  considerable  part  of  the  student's 
difficulty  in  grasping  the  matter  of  relative  accuracy 
lies  in  the  facts  that  he  has  usually  had  very  little 
practice  in  careful  measurement  and  that  his  previous 
study  of  arithmetic  has  emphasized  a  condition  of 
infinite  accuracy  of  numerical  values.  Such  a  number 


38         ACCURACY  AND  SIGNIFICANT  FIGURES 

as  12.5  is  supposed  not  only  to  mean  12.50,  but  also 
to  be  equal  to  12.500000 ...  to  an  unlimited  number 
of  decimal  places.  This  is  quite  proper  and  satisfactory 
as  long  as  one  realizes  that  he  is  dealing  with  imaginary 
quantities,  or  perhaps  it  would  be  better  to  speak  of 
them  as  ideal  quantities,  perfections  of  measurement 
which  have  no  more  real  existence  than  the  point,  line, 
plane,  or  cube,  of  the  geometrician.  The  smooth  sur- 
face of  a  table  does  not  come  as  near  to  being  a  plane 
as  does  the  surface  of  an  optically  worked  block  of 
glass  or  a  Whitworth  plane,  and  even  the  smoothest 
possible  surface  can  be  magnified  so  as  to  show  that 
it  contains  irregularities  everywhere.  Perhaps  if  it 
were  magnified  enough  we  could  see  that  its  shape  was 
not  even  constant,  but  individual  molecules  would  be 
found  swinging  back  and  forth  or  even  escaping  from 
the  surface.  A  geometrical  plane  certainly  corresponds 
to  nothing  in  reality,  outside  of  the  imagination,  and 
perfect  accuracy  of  number  is  just  as  much  an  imagi- 
nary concept. 

Decimal  Accuracy.— If  12.5  cm,  as  a  measurement, 
does  not  mean  12.50000 ...  to  an  infinite  number  of 
decimal  places  what  does  it  mean?  As  different  meas- 
urements are  often  made  with  different  degrees  of 
accuracy  the  universally  adopted  convention  is  merely 
the  common-sense  one  that  the  statement  of  a  meas- 
urement should  be  accurate  as  far  as  it  goes,  and 
should  go  far  enough  to  express  the  accuracy  of  the 
determination.  Thus  12.5  cm  means  that  a  length 
is  nearer  to  precisely  12.5  than  to  precisely  12.4  or 
12.6  cm;  12.50  cm,  however,  implies  that  the  stated 
length  is  nearer  the  exact  value  of  12.5  than  to  either 


ACCURACY  AND  SIGNIFICANT  FIGURES         39 

12.49  or  12.51,  that  is,  that  it  has  been  measured  to 
the  nearest  tenth  of  a  millimetre  and  does  not  differ 
from  twelve  and  a  half  centimetres  by  more  than  .005 
cm,  while   the  expression  12.5  cm  states  merely  that 
the    length   is   between    124J    and    125|    millimetres. 
If  the  length  had  been  12.5  cm  (i.  e.}  any  of  the  fol- 
lowing:   12.46,  12.47,  12.48,  12.49,  12.50,  12.51,  12.52, 
12.53,   12.54    cm)  the    statement    that    it   was  12.50 
cm  would  violate    the  requirement  that  a  measure- 
ment should  be  accurate  as  far  as  it  goes,  for  the 
chances  are  eight  to  one  that  it  would  be  one  of  the 
other  numbers  of  hundredths  given  above.     On  the 
other  hand,  if  an  observer  determined  a  length  to  be 

12.50  cm  and  only  stated  it  as  12.5  cm  he  would  not 
be  doing  justice  to  his  own  accuracy.     The  principle 
is  simple  enough  when  it  has  once  been  realized,  and 
then  there  is  little  danger  of  the  student's  ''rounding 
off"  a  carefully  obtained  measurement  like  2.638  gm. 
to  2 . 64  gm.  merely  for  the  sake  of  doing  some  rounding 
off;    but  he  will  probably  find  it  necessary  for  some 
length  of  time  to  be  careful  not  to  cut  off  a  final  sig- 
nificant zero.    If  the  brass  x-disc  is  found  to  be  "just 
8"  centimetres  in  diameter  the  measurement  should 
be  stated  as  8.00  cm  if  tenths  of   a  millimetre  were 
estimated  and  none  found;    but  it  should  be  given  as 
8.0  cm  if  the  student   read   the  millimetres  but  was 
unable  to  estimate  smaller  amounts.    There  is  nothing 
to  show  which  degree  of  accuracy  was  obtained  if 
the  diameter  is  put  down  as  8  cm  "because  it  came 
out  just  even." 

Significant  Figures. — The  figures  of  which  a  number 
is  composed,  if  they  are  all  necessary  to  express  its 


40         ACCURACY  AND  SIGNIFICANT  FIGURES 

accuracy,  are  called  significant  figures,  but  ciphers 
added  to  the  right  of  the  other  figures  merely  to  fill  out 
a  column  are  not  significant,  nor  are  ciphers  on  the 
left  of  a  number,  such  as  0.75  or  050.  Neither  are 
ciphers  significant  when  used  merely  to  locate  the 
position  of  the  decimal  point,  whether  on  the  left,  as 
in  the  statement  that  the  length  of  a  certain  light-wave 
is  .00005086  cm,  nor  on  the  right,  as  when  the  sun 
is  stated  to  be  93000000  miles  from  the  earth;  the 
first  of  these  numbers  has  four  significant  figures,  the 
second  only  two.  It  will  be  noticed  that  with  a  number 
like  the  last  there  is  trouble  in  applying  the  rule  of 
making  it  "accurate  as  far  as  it  goes."  Only  the  first 
two  or  three  figures  are  known,  but  eight  are  needed  in 
order  to  place  the  decimal  point.  Moreover,  suppose 
the  distance  should  be  found  to  be  93.000  millions  of 
miles  at  some  particular  time;  how  can  we  make  a 
distinction  between  93,000,ooo  with  five  significant 
figures,  and  93, 000,000  with  only  two,  unless  we  employ 
two  different  symbols  for  a  cipher,  one  to  be  used  when 
it  is  significant  and  the  other  when  it  is  not?  (The 
solution  of  the  difficulty  will  be  found  further  on  in 
this  lesson.) 

The  length  of  an  inch  has  been  determined  to  be 
between  25.3997  and  25.3998  mm.  How  many  of  the 
following  statements  are  correct  and  how  many  are 
positively  wrong? 

1  in.  =  25.4  mm.  1  in.  =25.40  mm. 

1  in.  =25.400  mm.  1  in.  =25.4000  mm. 

Which  of  the  following  values  of  IT  are  correct? 
3.141       3.142       3.1415       3.1416      3.141600       3.141592 


ACCURACY  AND  SIGNIFICANT  FIGURES        41 

Relative  Accuracy. — The  relative  accuracy  of  a 
number  depends  upon  two  things:  how  much  its 
absolute  difference  from  the  truth  amounts  to,  and 
how  large  the  number  itself  is.  If  two  points  on  the 
earth's  surface  are  found  by  careful  surveying  to  be  10 
miles  apart  the  determination  of  distance  may  easily 
be  in  error  by  more  than  a  foot,  and  even  with  the 
most  extremely  careful  triangulation  the  error  is  likely 
to  be  as  much  as  four  inches.  An  error  of  a  quarter 
of  an  inch,  however,  in  measuring  the  thickness  of 
a  door  could  hardly  be  made  even  with  the  clumsiest 
of  measuring  apparatus.  It  is  plainly  misleading  to 
say  that  the  latter  determination  is  more  accurate 
than  the  former  because  J  in.  is  less  than  four  inches. 
The  size  of  the  error  means  nothing  unless  we  also 
know  the  size  of  the  measurement  in  which  it  occurs. 
Suppose  the  thickness  of  the  door  is  1^  inches,  then 
the  error  amounts  to  one  sixth  of  the  whole  measure- 
ment, or  is  an  error  of  about  16%.  An  error  of  four 
inches  out  of  ten  miles,  however,  is  roughly  an  error 
of  one  out  of  a  hundred  and  fifty  thousand,  or  about 
six  per  million,  or  about  .0006  of  1%. 

Calculation  of  Relative  Errors. — The  relative  error  of 
a  measurement  is  not  usually  needed  with  very  great 
accuracy.  Where  numbers  are  as  different  as  6  in 
tens-of-thousandths'  place  and  16  in  units'  place  the 
location  of  the  decimal  point  is  the  only  thing  of 
importance  and  the  numbers  in  the  corresponding 
situations  can  almost  be  ignored.  Consequently,  a 
rough  calculation  of  a  relative  error  is  all  that  is  neces- 
sary, and  this  can  be  performed  mentally.  Thus,  in 
the  above  example,  1  foot  is  1/5280  of  1  mile,  hence 


42         ACCURACY  AND  SIGNIFICANT  FIGURES 

it  is  about  1/50,000  of  10  miles;  and  4  inches,  being 
J  of  1  ft.,  is  about  l/(3  X  50,000),  or  1/150,000.  Fif- 
teen is  about  J  of  100,  so  150,000  is  about  J  of  1,000,000; 
and  1/150,000  =  6/1,000,000  or  .000006  or  .0006%. 

If  a  length  is  stated  as  174.2  cm.  the  inference  is 
that  it  is  nearer  to  that  exact  amount  than  to  174.1  or 
174.3,  namely,  that  its  error  certainly  is  not  as  much 
as  0.1  out  of  174.2;  this  means  that  it  is  not  as  much 
as  1  out  of  1742,  or  1  out  of  nearly  2000,  or  5/10000,  or 
.0005,  of  .05%. 

Copy  the  following  numbers  and  write  beside  each 
one  its  approximate  accuracy;  e.  g.}  66.7  is  stated 

with  an  accuracy  of  0.1  out  of  a  total 

7.23  of  66.7,  or  of  one  part  in  667,  or 

90  512     about  ^  per  1000'  or  -001*'  or  >15//0> 
428.          Which  column  has  its  numbers  stated 


7.23 
94.07 

0.51 

428.00 

66.67 


•'        with  about   the  same  degree  of  ac- 


curacy, the  one  in  which  each  number 
is  given  with  the  same  number  of  decimal  places,  or  the 
one  in  which  each  has  the  same  number  of  significant 
figures? 

What  was  the  relative  error  of  your  measurement  of 
the  width  of  the  table  by  spans  and  finger-breadths 
in  Lesson  I? 

In  Lesson  II,  when  the  relative  error  of  a  measured 
sine  was  determined  did  you  pick  out  the  largest 
relative  error  by  choosing  the  largest  numerical  error? 

Rule  for  the  Relative  Difference  of  Two  Measure- 
ments.— When  the  theoretical  value  of  a  number  is 
known  the  numerical  error  is  divided  by  the  true  value, 
as  we  have  seen.  When  there  is  no  theoretical  standard 
and  there  are  simply  two  experimental  determinations 


ACCURACY  AND  SIGNIFICANT  FIGURES         43 

to  be  compared  with  each  other  the  accepted  pro- 
cedure is  to  divide  the  difference  by  the  greater  value. 
Apply  this  rule  to  your  two  measurements  of  an 
irregular  area  in  Lesson  I,  obtained  by  counting  squares 
and  by  constructing  geometrical  figures.  How  much 
relative  difference  is  there  in  the  results  of  the  two 
methods. 

Calculate  719  X  327  and  719  X  325;  if  the  third 
figure  of  one  factor  changes  how  many  figures  of  the 
product  can  be  relied  upon  to  remain  unchanged? 
Compare  the  last  result  with  the  example  on  page  18; 
if  the  factors  are  stated  correctly  to  three  figures  how 
many  figures  of  the  product  will  be  correct?  Decide 
mentally,  by  induction,  how  many  figures  of  a  product 
can  be  trusted  if  one  of  its  factors  contains  six  figures 
and  the  other  one  has  only  three. 

Remembering  that  the  accuracy  with  which  a  quan- 
tity is  expressed  depends  not  upon  the  number  of 
decimal  places  but  upon  the  number  of  significant 
figures  and  keeping  in  mind  the  fact  that  the  number 
of  trustworthy  figures  in  a  product  is  the  same  as  the 
number  in  its  least  accurate  factor,  turn  back  to  your 
notes  on  physical  arithmetic  and  observe  that  the 
method  of  abridged  division  automatically  gives  just 
the  number  of  figures  in  the  quotient  that  are  needed 
if  no  figures  of  the  dividend  are  " brought  down"; 
and  that  abridged  multiplication  always  gives  at  least 
as  many  as  are  in  the  smallest  factor.  It  will  not  give 
any  superfluous  figures  if  the  larger  factor  is  used  as 
the  multiplier,  but  will  give  as  many  as  the  larger 
factor  contains  if  that  is  used  as  the  multiplicand. 


44         ACCURACY  AND  SIGNIFICANT  FIGURES 

Of  course  the  best  method  is  to  round  off  the  larger 
factor  before  multiplying,  so  that  it  has  no  more  figures 
than  the  smaller  one. 

Standard  Form. — To  avoid  a  long  string  of  figures 
when  writing  very  large  or  very  small  numbers  it 
is  customary  to  divide  a  number  into  two  factors, 
one  of  them  a  power  of  ten.  Thus  .000000017  and 
632000000000  are  the  same  as  17  X  lO^9  and  632  X  109 
respectively.  This  notation  also  makes  it  possible  to 
write  93000000  unequivocally  with  two  significant  fig- 
ures or  with  four,  as  may  be  desired,  for  we  can  put  it 
either  in  the  form  9.3  X  107  or  in  the  form  9.300  X  107. 
The  same  value  and  accuracy  for  the  second  of  these 
would  be  retained  just  as  well  by  writing  93.00  X  106 
or  930.0  X  105,  but  it  is  customary  to  choose  the  p.ower 
of  ten  so  that  the  other  factor  shall  have  just  one 
significant  figure  to  the  left  of  the  decimal  point. 
The  number  is  then  said  to  be  written  in  standard  form. 

Write  the  following  numbers  in  standard  form: 

2946.3;  0.051  (Ans:  5.1  X  1Q-2):  666.6;  .0004;  18.27;  17.042; 
1.12;  25.4;  25.400;  186000. 

Write  a  definition  of  standard  form  in  your  own 
words. 


LOGARITHMS  45 

IV.   LOGARITHMS 

In  the  following  equations  the  constant  10  is  called 
a  "base"  and  any  exponent  is  called  the  " logarithm" 
of  the  right-hand  side  of  the  equation;  10s  =  looooo 
thus,  3  is  said  to  be  the  logarithm  of  1000,  104  =  10000 

or,  as  it  is  usually  abbreviated:  103    =  1000 

102    =  100 
101    =  10 

3  =  log  1000.  10°    =  1 

10-1  =     .  1 
10-2  =      oi 
10~3  =     .001 

Properties  of  Logarithms. — Those  logarithms  whose 
base  is  10  are  called  " common"  logarithms.  For  theo- 
retical purposes  "natural"  logarithms  are  sometimes 
used;  their  base  is  "e,"  approximately  2.7,  or  1  +  1  + 
1/2  +  1/2.3  +  1/2.3.4  +  1/2.3.4.5  +  .  .  .  Notice  from  the 
equations  given  that  log  100  +  log  1000  =  log  100,000. 
This  suggests  the  general  relationship  that 
log  a  +  log  b  =  log  (a  X  6) 

Try  numerical  values  for  the  following  also: 
log  a  —  log  b  =  log  (a/6) 
n  X  log  a  =  log  (an) 
(log  a}  fn  =  log  jya 

These  four  equations  express  the  fundamental  prin- 
ciples that  in  the  use  of  logarithms  "addition  takes 
the  place  of  multiplication,  subtraction  of  division, 
multiplication  of  raising  to  a  power,  and  division  of 
root  extraction." 

The  advantage  of  using  10  for  a  base  is  that 

log  (10  X  a)  =  log  10  +  log  a  =  1  +  log  a 


46  LOGARITHMS 

and  in  general 

log  (10nX  a)  =  n  log  10  +  log  a  =n  +  log  a; 

for  example,  log  365  =  2  +  log  3.65.  Accordingly 
tables  of  common  logarithms  are  made  out  only  for 
numbers  between  1  and  10,  the  logarithms  of  all  other 
numbers  being  self-evident  from  these. 

The  logarithms  of  numbers  other  than  powers  of 
10  are  in  general  incommensurable  and  are  given  only 
approximately  in  tables.  Using  the  small  table  given 
here  find  2X3.  Answer:  log  2  =  .301;  log  3  =  .477; 
their  sum,  .778,  is  found  from  the  table  to  be  log  6, 
or,  as  it  is  said,  the  antilogarithm  of  .778 


no. 


log.  is  6.  Find  2X4;  22;  32;  4X5;  V9;  5 
~^  X  6;  50  X  6;  50  X  600.  Calculate  the  value 
.301  of  e  from  the  series  on  page  45.  Examine 

•  g^     the  four-place  logarithm  table  on  pages  142 
^699     and  143,  and  notice  that   it    contains  the 
•j^     same  succession  of  numbers,  from   1.00  to 
i903     9.99,  and  of  logarithms,  from  .0000  to  .9999, 

•  954     as  m  the  table  on  this  page,  but  with  in- 

termediate values  at  smaller  intervals,  and 
without  any  decimal  points.  Verify  the  following  state- 
ments by  finding  the  logarithm  in  line  with  the  first 
two  figures  in  the  left-hand  column  and  in  the  column 
headed  by  the  third  figure: 

log  3.65  =  .5623;  log  3.66  =  .5635;  log  4.06  =  .6085;  log 
7.70  =  .8865;  log  77.0  =  1.8865;  log  0.77  -  -  1.  +  .8865; 
log  .00077_=  —  4.  +  .8865,  or,  as  it  is  generally  written,  log 
.00077  =  4.8865,  the  minus  sign  being  written  over  the  4  to 
indicate  that  it  applies  only  to  the  whole  number,  and  the  deci- 
mal part,  or  "mantissa"  as  it  is  sometimes  called,  being  always 
positive. 


LOGARITHMS  47 

Write  the  integral  part,  or  "characteristic,"  of  the 
logarithm  of  each  of  the  following:  5441  (Ans.:  3); 
27;  79264;  264;  73;  0.73;  0.073;  0.000073.  Make 
up  a  rule  for  finding  the  characteristic  of  any  logarithm. 

Write  the  logarithms  of  984;  982;  981;  980;  98; 
9.8;  .98;  .098;  7;  14.  Add  the  last  two  and  find  the 
result  in  the  body  of  the  table ;  see  what  marginal  num- 
ber is  opposite  it,  and  verify  by  multiplying  14  by  7. 

Use  of  the  Table  of  Logarithms. — A  four-place  table 
is  in  general  satisfactory  for  obtaining  the  logarithm 
of  a  number  that  has  as  many  as  four  significant 
figures;  for  five-figure  accuracy  a  five-place  table  is 
necessary,  etc.  The  process  of  finding  the  logarithms 
of  numbers  lying  between  two  consecutive  tabular 
numbers  will  be  easily  understood  from  the  following 
example:  Find  \/3.142~  Log  314  ==  4969;  log  315  = 
4983;  difference  =  14;  a  number  that  is  two  tenths 
of  the  way  from  314  to  315  will  have  a  logarithm  that 
is  two  tenths  of  the  way  from  4969  to  4983;  0.2  of 
14  =  3  (to  the  nearest  integral  value) ;  4969  -f  3  = 
4972.  Log  3.142  =  .4972;  .'*.  log  \/JU42  =  .4972  -*- 
2  =  .2486.  The  logarithm  2486  does  not  occur  in  the 
table  but  is  between  2480  ( =  log  177)  and  2504  ( =  log 
178) ;  2486  is  6/24  of  the  way  from  2480  to  2504,  and 
hence  is  the  logarithm  of  177^4  or  17725  (better  say 
1772,  as  the  fifth  figure  is  liable  to  be  incorrect).  Ans. : 
1.772. 

Notice  that  the  small  multiplication  tables  at  the 
side  of  the  main  table  enable  the  multiplications  and 
divisions  to  be  performed  mentally. 

Practice  finding  reciprocals  mentally  by  the  process 
illustrated  in  the  following  examples:  Required,  the 


48 


LOGARITHMS 


value  of  1  •*•  3.142.  Log 3. 142  =  .4972;  0  -.4972  =  .5028, 
subtracting,  from  left  to  right,  each  figure  from  9, 
except  the  last,  which  is  subtracted  from  10;  .5028  = 
log  3183;  answer,  pointed  off  by  inspection,  .3183.  The 
value  of  log  1  —  log  x,  or  0  —  log  x,  is  known  as  the  co- 
logarithm  of  x. 

Satisfy  yourself  that  the  following  reasoning  is  correct : 
If  we  assume  that 

y  =  e-*2 
then,  taking  logarithms  of  each  side, 

logy  =  -  x*  log  e 
and 

—  logy  =  x2  log  e', 

taking  logarithms  again 

log  (-  log  y)  =  log  (x2)  +  log(log  e) 
or 

log  (-  log  y}  =  log  x2  +  1.6378. 

Using  the  last  equation  and  logarithm  tables  find 
x2,  log  x2,  log  x~  +  1.6378,  log  (-log  y),  log  y,  and  y 
for  each  of  the  following  values  of  x:  0,  .2,  .4,  .6,  .8, 
1.0, 2.6,  2.8,  3.0,  4,  5.  On  the  next  unused  left- 
hand  page  of  your  note-book  tabulate  the  results  in 
columns  headed  x,  x2,  log  x2,  etc.  Leave  the  opposite 
right-hand  page  vacant  until  after  studying  graphic 
representation. 


X 

0 

x2 
0 

logx* 

Iogx2  +  1.6378 

log  (-logy) 

-logy 

+  logy 

y 

—  00 

—  00 

—  GO 

0 

0 

1 

.2 

.04 

2.6021 

2.2399 

2.2399 

.0174 

1.9826 

.961 

.4 
.6 

.16 

1  .  2041 

2.8419 

2.8419 

.0695 

1  .  9305 

.852 

3^0 
4.0 
5.0 

SMALL  MAGNITUDES  49 


V.  SMALL  MAGNITUDES 

Apparatus. — Platform  balance;  set  of  weights  from 
1  gm  to  500  gm,  set  of  avoirdupois  weights  from  1  oz. 
to  8  oz. 

Negligible  Magnitudes. — The  way  in  which  small 
magnitudes  enter  into  physical  calculations  may  be 
seen  in  the  following  example : 

Suppose  that  a  metal  cube  has  been  constructed 
accurately  enough  to  measure  1.00000  cm  along  each 
edge.  If  it  was  brought  from  a  cold  room  into  a 
warm  room  a  delicate  measuring  instrument  might 
show  that  the  change  of  temperature  had  increased 
each  dimension  to  1.00012  cm,  and  by  unabridged 
multiplication  it  would  be  easy  to  prove  that  the 
area  of  each  side  was  1.0002400144  cm2  and  that 
the  volume  had  become  1.000360043201728  cm3.  If 
the  most  careful  measurements  just  allow  us  to  dis- 
tinguish units  in  the  fifth  decimal  place  then  tenths 
of  those  units  (represented  by  the  sixth  decimal  place) 
would  be  impossible  to  measure,  and  the  attempt  to 
state  not  only  tenths,  but  hundredths  and  thousandths 
of  those  units  becomes  absurd.  By  noticing  that  the 
number  1.0002400144  differs  from  the  value  1.00024 
that  would  be  obtained  by  abridged  multiplication 
only  as  much  as  one  or  two  thousandths  of  the  smallest 
measurable  amount  we  can  see  clearly  why  the  area 
of  a  1.00012-cm  square  is  and  must  be  1.00024  cm2. 
Similarly,  the  volume  of  the  cube  is  neither  more  nor 
less* than  1.00036  cm3,  and  the  string  of  figures  running 
out  ten  decimal  places  further  is  absolutely  meaningless. 


50  SMALL  MAGNITUDES 

It  will  be  noticed  that  1.0002400144  is  in  the  same 
form  as  1  +  2x  -f  x2,  the  square  of  (1  +  x),  where 
x  =  .00012;  also,  that  1.000360043201728  corresponds 
to  1  +.3x  +  3z2  +  z3,  the  cube  of  (1  +  x).  In  other 
words,  when  dealing  with  the  objects  of  the  real 
world  which  is  evident  to  our  senses  it  may  happen 
that  a  measured  amount  is  so  small  that  its  higher 
powers,  algebraically  speaking,  are  minute  beyond  all 
perception.  Of  course  this  must  not  be  understood 
as  meaning  that  the  cube  of  a  measurable  length  can 
ever  be  an  impalpable  volume;  the  cube  of  1  +  x  is 
even  a  larger  number,  1  +  3z;  it  is  the  difference 
between  this  "physical"  value,  (1  +  x)3  =  1  +  3x,  and 
the  mathematical  value,  (1  +  z)3  =  1  +  3x  +  3x2  +  x3 
that  eludes  perception  on  account  of  x2  being  extremely 
small  in  comparison  with  x,  which  is  itself  minute. 

The  examples  that  have  been  given  above  suggest 
that  if  x  is  small  enough  (1  +  x)2  =  1  +  2x,  (1  +  x)3  = 
1  +  3x,  and  in  general  (1  +  x)n  =  1  -f  nx.  The  mat- 
ter can  be  tested  by  making  use  of  the  binomial  theorem, 
that 

n(n-l)  n(n-l)(n-2) 

(1  *  xy  =  j  ±  nx  +  ..  Si  >2     3.2  ±          j  .2.3   -    *3  +  .  •  - 

This  shows  that  (1  ±  x}n  =  1  =*=  nx  if  x  is  so  small 
that  x2,  x*,  etc.  are  negligible,  the  only  possible  excep- 
tion being  in  case  n  should  be  so  large  that  it  could 
counterbalance  the  small  size  of  x  and  prevent  the 
term 


from  becoming  negligible.    In  physical  measurements, 
however,  n  is  never  large  enough  for  this  to  happen. 


SMALL  MAGNITUDES  51 

Properties  of  Deltas. — The  small  quantities  which 
we  have  been  considering  are  usually  denoted  by  the 
Greek  letter  5  (read  " delta"),  and  one  of  the  most 
important  properties  of  deltas  is 

(1  -j-  a)»  =  1  +  nd. 

It  is  advisable  to  learn  such  an  equation  in  the  form 
given  here  rather  than  in  the  equivalent  form  (1  ±  5)n 
=  1  =*=  nd,  on  account  of  possible  confusion  in  applying 
equations  containing  more  than  one  double  sign.  Of 
course  6  can  be  considered  as  having  a  negative  value 
when  necessary. 

Find  the  square  of  0.97.     Ans:     0.97  =  1.  -  .03; 
=  1  +  (-  .03);    0.972  =  1  +  2  (-  .03)  =  1.  -  .06  = 
.94,  correct  to  as  many  significant  figures  as  are  given. 
Find  (1.00012)4  mentally. 
Find  (.99988)4  mentally. 
By  algebraical  division: 

1/(1  +  x)  =  1  -  x  +  x2  -  x3  +  x4  -  . 
whence 

"I  t 

1  +8  = 

Divide  1  by  .995.  Ans:  1/(1.  -  .005)  = 

1.  -  (-  .005)  =  1.005. 

Find  mentally  the  reciprocal  of  1.00012. 

Find  1/(1.00012)2  by  using  first  one  formula  and  then 
the  other. 

Find  (1.00012)1'2,  and  complete  the  following 
formula: 

SAT»- 


52  SMALL  MAGNITUDES 

By  ordinary  multiplication  (1  =*=  x)  (1  =±=  y)  =  1  =1=  x 
±  y  -i-  xy,  and  if  both  x  and  ?/  are  small  their  product 
xy  will  be  negligible,  so  that 

(1  +  5:)  (1  +  S2)  =  1  +  Si  +  52 

Find  1.00012  X  .99890.  Ans: 

(1  +  .00012)  (1  +  (-  .00110))  =  1  +  .00012  -  .00110 
=  1.  -  .00098  =  .99902. 

Find  (1.0021)  (1.0037)  in  three  different  ways,  writing 
out  the  complete  work  in  your  note-book:  (a)  by  ordi- 
nary multiplication,  (b)  by  abridged  multiplication, 
(c)  by  the  use  of  deltas. 

A  barometer  reading  is  corrected  for  temperature 
errors  by  multiplying  it  by  1.00037  and  dividing  by 
1.00364;  what  is  the  percentage  difference  between  the 
corrected  reading  and  the  original  reading?  Suggestion : 
call  the  original  height  unity. 

Write  the  formula  for  (1  +  Si) /(I  +  52). 

If  two  numbers  are  nearly  equal  they  may  be  denoted 
by  a  and  a  +  5,  to  indicate  the  fact  that  their  difference 
is  a  small  magnitude.  Then 

\/a(a  +  6)  =  \/a-a(l  +  5/a)  =  a\/l  +  5/a  =  a(l  +  S/2o) 
since  5/a  is  also  a  small  magnitude.    But 

o(l  +  8/2o)  =  a  (2o  +  5)/2a  =  -     ±-j~- 

In  other  words,  if  two  quantities  are  nearly  equal 
the  square  root  of  their  product  (or  geometrical  mean) 
can  be  found  by  taking  their  average  (or  arithmetical 
mean) . 

If  an  object  has  an  apparent  weight  of  m\t  when 


SMALL  MAGNITUDES  53 

placed  on  one  pan  of  a  balance,  and  m2  when  on  the 
other  pan,  it  can  be  proved  that  its  true  mass  is  \Xmim2. 
Weigh  your  whole  set  (16  oz.)  of  avoirdupois  weights, 
considered  as  an  unknown  mass,  on  the  platform  bal- 
ance against  the  brass  metric  weights.  Repeat  the 
process  on  the  other  pan  of  the  balance.  Find  the 
true  mass  of  the  avoirdupois  set  in  grams  from  the 

formula  : 

(<*)  +  (a  +  5) 


Draw  a  diagram  similar  to  the  one  on  page  30,  but 
with  a  very  small  angle  at  O.  It  will  be  almost  self- 
evident  that 

tan  5=5 
and 

sin  5=5, 

remembering  that  the  angle  8  is  to  be  measured  as  the 
ratio  of  arc  to  radius. 

By  consulting  the  table  of  circular  functions  on  page 
141,  find  the  largest  whole  number  of  degrees  for  which 
tan  8  =  3  to  four  decimal  places.  The  numerical 
measure  of  each  angle  will  be  found  beside  the  number 
of  degrees  in  the  column  headed  RAD,  an  abbreviation 
of  radian,  the  name  sometimes  given  to  the  unit  of 
angle.  For  how  large  an  angle  is  it  true  that  sin  8  =  8 
as  far  as  four  decimal  places? 

If  an  accuracy  of  three  decimal  places  is  all  that  is 
needed  how  large  can  6  be  without  differing  from  tan  8 
and  sin  8  respectively? 

Recapitulation.  —  The  formulae  for  deltas  are  collected 
for  reference  here.  Notice  that  the  second  and  third 
are  special  cases  of  the  first,  and  the  first  formula 


54  THE  SLIDE  RULE 

for  n  equal  to  a  whole  number  only,  is  a  special  case  of 

the  fifth. 

(1  +  S)»  =  1  +  n& 
•  1/(1  +  5)  =  1  -  6 


(1  +  5.)  (1  +  52)  (1  +  53)  .  .  .   =  1  +  5:  +  52  +  53  .  .  . 


/an  5  =  sin  5  =  5 

Find   V50  by  using  the  5-formulse. 
Answer:  V  50  =  V49—  1  =  V49(l  -  1/49)  =  7  Vl  ^1/49  = 


7  (1  +  1/98)   =  7   (l  +  105=2)  =  7  (l  +  I^(1  +  .02))  = 
7  (1.0102)  =  7.0714. 

The  first  four  figures  of  this  result  are  correct. 
Extreme  accuracy  cannot  be  expected  where  a  delta 
is  larger  than  .005;  in  most  physical  calculations  it  is 
much  smaller  than  this. 

Find  400  -f-  797.  Suggestion:  divide  through  by  800 
in  order  to  put  the  divisor  in  the  form  1  +  6. 

Find  .504  X  .498. 

VI.   THE  SLIDE  RULE 

Apparatus.  —  A  ten-inch  slide  rule,  with  A,  B,  C, 
D,  L,  S,  and  T  scales,  a  runner,  and  a  list  of  metric 
equivalents. 

Addition  with  Two  Scales.—  If  two  scales  of  centi- 
metres are  laid  parallel  so  that  the  zero  of  the  second 
one  coincides  with  the  seventh  division  of  the  first 


THE  SLIDE  RULE  55 

it  is  evident  that  the  fifth  division  of  the  second  will 
be  opposite  the  twelfth  division  of  the  first.  A  length 
of  5  has  been  added  to  a  length  of  7,  and  the  result  is 
seen  at  a  glance  to  be  12.  It  will  also  be  noticed  that 
the  arithmetical  difference  between  scale  divisions 
that  lie  opposite  each  other  is  everywhere  the  same 
and  is  equal  to  the  number  on  the  first  scale  which  is 
opposite  the  beginning  of  the  second. 


12345678 

i        i        i        i        i        i                i 

9 

j 

10  11 

i  i 

12  fd 

i  i 

i 

1 

2 

i 

3  4 

i  i 

5  6 

i  i 

7 

i 

Multiplication  with  Logarithmic  Scales.— If  the  same 
experiment  is  tried  with  two  scales  whose  divisions  are 
not  a  succession  of  whole  numbers  but  the  logarithms 
of  such  a  series  the  result  will  be  different,  for  adding 
logarithms  corresponds  to  multiplying  natural  num- 
bers. Accordingly  if  we  start  at  log  7  and  measure 
a  further  distance  of  log  5  we  shall  come  out  with 


1       1.5    2 

,,,,1,,,,! 

3 

1 

4    5 

,1,1 

6-J 

.1, 

r8S 

,1. 

K)         15    20 

,1  1  1  J    Illlll 

30 

,  | 

40       60 

,   1  ,  M 

80 

,h 

100 

[,L, 

150 

M|.M 

f 

1  1  ,  i  n 

1.5  ; 

L3 

'  1  ' 
4 

1  '  I'l'l'l'    " 
5  6  78910 

^PI 

15 

20  25 

log  7  +  log  5,  which  is  not  equal  to  log  (7  +  5)  but  to 
log  (7X5).  Notice  that  not  only  does  five  on  the 
second  scale  come  opposite  35  on  the  first,  but  also 
that  quotients  of  corresponding  numbers  are  every- 
where equal  to  7,  just  as  differences  are  in  the  first 
diagram;  and  furthermore,  the  upper  scale  in  the 
second  diagram  forms  a  multiplication  table  (a  7  table 


56  THE  SLIDE  RULE 

in  this  case)  for  the  numbers  on  the  lower  scale  just 
as  it  did  an  addition  table  in  the  other  diagram. 

The  Slide  Rule. — The  apparatus  called  a  slide  rule 
is  essentially  a  ruler  containing  a  groove  in  which  is  a 
movable  slide.  Logarithmic  scales  are  so  marked  that 
one  of  them  (the  "A  scale")  can  be  held  stationary 
while  another  (the  "B  scale")  can  be  placed  in  any 
required  position  below  it. 


f.5     2         3V    4     5   6  78910         15     20        30     40   60  60 70 ,80 Jo fOO 

il.Mil      .    it  .   I  .  I  .  I    I   Mini  ilniil      I     I    .   I   ,  I         I  it  i  I 


1  yw        a  .      3'     4    s    e  7  g 

C       >•'.',    '    ,',  I  ,'    '    '    '    i. i  T         if      '|        i      7      .     "j     ;  Tl    '   "i    * 

i  2  3  456789    10 


Two  scales  (C  and  D)  along  the  lower  edge  of  the 
slide  can  be  used  in  the  same  way  and  can  be  read 
a  little  more  accurately  on  account  of  their  sub- 
divisions being  larger,  but  the  two  upper  scales  (A 
and  B)  will  be  found  the  most  convenient  for  gen- 
eral use.  Each  one  of  them  is  really  two  complete 
logarithmic  scales  from  1  to  10,  but  the  right-hand 
half  is  often  marked  10  to  100.  If  the  numbers  are 
the  same  on  both  halves  it  is  necessary  to  remember 
that  any  graduation,  such  as  7,  can  be  used  to  represent 
7,  or  70,  or  7000,  or  .07,  as  may  be  required.  Care 
should  be  taken  to  avoid  mistakes  in  reading  the 
graduations  between  the  whole  numbers,  for  the 
smallest  interval  is  not  everywhere  the  same,  being 
.1,  .05,  .02,  and  .01  in  the  various  parts  of  the  scales. 
Directions  have  been  worked  out  for  finding  the  posi- 
tion of  the  decimal  point  in  a  result  obtained  by  means 


THE  SLIDE  RULE  57 

of  the  slide  rule,  but  it  is  preferable  to  form  the  habit 
of  always  determining  the  approximate  answer  before 
making  any  arithmetical  calculation,  whether  it  is 
made  with  the  slide  rule  or  not. 

Checking  by  Approximation. — This  is  one  of  the  best 
methods  of  checking  the  accuracy  of  the  result  and 
guarding  against  ridiculous  mistakes.  For  example, 
suppose  23.4  is  to  be  multiplied  by  82.9;  the  computer 
should  say  that  20  X  80  =  1600,  .'.  23.4  X  82.9  must 
be  somewhat  larger  than  1600.  When  the  answer  is 
found  by  the  slide  rule  to  have  the  significant  figures 
194  there  can  be  no  doubt  that  it  should  be  pointed  off 
1940,  not  19.4  or  19400.  Furthermore  if  the  answer 
obtained  either  with  the  slide  rule  or  by  abridged  mul- 
tiplication or  by  any  other  method  should  come  out 
613  it  would  be  evident  that  a  mistake  had  occurred 
somewhere. 

Multiplication. — To  find  the  product  of  two  numbers 
with  the  slide  rule  either  end  of  the  B  scale  should  be 
set  opposite  one  factor  on  the  A  scale,  then  the  other 
factor  on  the  B  scale  will  be  opposite  the  product  on 
the  A  scale.  Try  this  process  with  small  numbers  by 
setting  1  on  B  opposite  3  on  A  and  observing  that  the 
position  of  2  on  B  gives  2  X  3  on  A. 

Multiply  2  X  4;  2  X  5;  2  X  6;  3  X  9;  7  X  8;  7  X 
13;  7  X  16. 

Remember  that  the  two  halves  of  each  scale,  are 
identical.  If  16  (or  1.6)  when  taken  on  the  right  half 
of  the  B  scale  falls  beyond  the  end  of  the  A  scale  the 
same  result  can  be  read  over  16  (or  1.6)  on  the  left 
half. 

Multiply  1.5  by  2;  1.8  by  2;  1.7  by  2.1  (estimate 
third  figure  of  product);  17.8  by  2;  1.79  by  2.53. 


58  THE  SLIDE  RULE 

Find  0.6  X  183.;  7.3  X  1.09;  0.073  X  0.0016;  325. 
X  106.5. 

The  use  of  the  " standard  form"  often  makes  the 
preliminary  checking  easier;  thus  in  the  last  two 
examples  given,  (7+)  X  W~2  multiplied  by  (2—)  X 
10-3  gives  14  X  10~5,  or  .00014;  and  3  X  102  multi- 
plied by  1  X  102  gives  3  X  104  or  30000. 

Division. — In  the  first  diagram,  above,  it  may  be 
considered  that  a  length  of  12  is  laid  off  from  left  to 
right,  and  then,  beginning  at  the  point  12,  a  length 
of  5  is  laid  off  to  the  left,  or  subtracted  from  the  original 
12,  giving  a  final  result  of  7.  Similarly  in  the  second 
diagram  log  35  minus  log  5  equals  log  7.  The  rule  for 
division  is  accordingly:  place  the  divisor  on  B  under 
the  dividend  on  A  and  read  the  answer  on  A  over 
either  end  of  the  B  scale. 

Divide  35  by  5;  30  by  5;  30  by  4;  11  by  4;  11  by  7; 
11  by  10;  11  by  11;  11  by  12;  11.8  by  99.;  114.  by  3.4. 

Ratio  and  Proportion. — In  the  first  diagram  8^1  = 
9  —  2  =  10  —  3  =  7.  Since  subtraction  of  logarithms 
corresponds  to  division  of  natural  numbers  the  second 
diagram  shows  14/2  =  28/3  =  35/5,  etc.  That  is,  with 
the  slide  rule  set  in  a  given  position  any  two  opposite 
numbers  are  in  the  same  ratio  as  any  other  two.  Set 
6  under  2  and  notice  that  15  is  under  5.  Solve  the 
following  proportions  by  setting  the  rule  so  .that  the 
answer  is  always  found  on  the  A  scale. 

6  :  2  : :  15  :  x; 

3  :  2  : :  9  :  x  : :  12  :  y  : :  10  :  z; 
31  :  750  : :  .005  :  x 

(First  say  750  is  about  twenty  times  as  large  as  31.) 


THE  SLI&E  RULE  59 

Solve  the  following  equation  as  accurately  as  possible : 
26  :  66  : :  1  :  x 

If  26  inches  =  66  centimetres  what  is  the  length  of 
one  inch?  What  is  the  approximate  length  of  4  inches? 
How  many  centimetres  in  41  inches?  Turn  to  the  back 
of  the  slide  rule  and  see  what  statement  you  can  find 
of  the  relation  between  centimetres  and  inches.  Look 
for  a  similar  statement  in  regard  to  pounds  and  kilo- 
grams and  calculate  your  own  weight  in  kilograms, 
remembering  to  set  the  slide  rule  so  that  the  answer  is 
always  found  on  the  A  scale. 

Reciprocals. — Set  any  number  on  B  under  1  on  A; 
what  do  you  read  on  A  over  1  (or  10,  or  100)  on  B? 
Set  any  number  on  C  over  1  or  10  on  D;  what  do  you 
read  on  D  under  1  or  10  on  C?  What  advantage  have 
the  C  and  D  scales  as  compared  with  the  A  and  B  scales? 

Try  multiplication,  division  and  proportion  on  the 
C  and  D  scales,  remembering  to  set  the  slide  so  that 
the  answer  always  comes  on  the  stationary  (D)  scale. 
In  multiplication  if  the  answer  runs  off  the  end  of  the 
rule  re-set  the  slide  with  10  instead  of  1  on  C  opposite 
the  first  factor,  and  in  division  read  the  result  under 
10  instead  of  under  1  when  necessary.  If  the  fourth 
term  of  a  proportion  cannot  be  read  the  end  of  the  C 
scale  which  is  over  the  D  scale  must  be  precisely 
replaced  by  its  other  end. 

Set  26  over  66  and  find  the  number  of  centimetres 
in  five  inches.  After  setting  the  slide  move  the  trans- 
parent runner  so  that  its  vertical  black  line  exactly 
covers  1  on  C,  and  without  allowing  its  position  to 
shift  set  10  on  C  directly  under  it.  Then  under  5  on 
C  read  the  answer  on  D. 


60  THE  SLIDE  RULE 

Squares  and  Square  Roots. — Set  the  slide  so  that 
1  on  C  is  just  opposite  1  on  D,  and  move  the  trans- 
parent runner  so  that  its  vertical  line  falls  on  4  of 
the  A  scale.  Where  does  it  cut  C  and  D?  Set  it  suc- 
cessively at  9,  16,  25,  36,  and  49  on  A,  and  read  D. 
Set  it  at  9,  8,  and  7  on  D  and  read  A. 

The  square  of  any  number  is  given  unequivocally 
(except  for  the  location  of  the  decimal  point,  which  is 
easily  determined  by  inspection)  on  A  above  the  num- 
ber itself  on  D.  Care,  however,  is  necessary  in  revers- 
ing the  process,  for  a  given  arrangement  of  significant 
figures  has  two  different  square  roots,  as  is  shown  in 
the  following  equations: 

V  1500  =  40  -;  V  150  =  12  +;  V  15  =  4  -;  V  1.5  = 
1.2  +  ;  V  .15  =  .4  -;  V  .015  =  .12  +;  V  .0015  =  .04  -; 
V  -00015  =  .012  +. 

One  of  these,  121,  will  be  found  under  the  150  of 
the  left  half  of  the  A  scale,  the  other,  383,  under  the 
150  of  the  right  half.  The  simple  precaution  of  making 
a  rough  mental  preliminary  calculation  of  the  root 
will  avoid  the  possibility  of  using  the  wrong  number. 
For  example,  is  the  square  root  of  .036  given  by  the 
significant  figures  19  or  60,  and  how  should  it  be  pointed 
off? 

There  are  several  slightly  more  complicated  forms 
of  problem  which  can  be  solved  with  a  single  setting 
of  the  slide  rule.  Thus  using  the  upper  scales  we  can 
not  only  read  the  value  of  a/6  by  setting  b  under  a, 
but  we  can  also  use  the  method  of  slide-rule  multipli- 
cation to  give  the  value  of  |  X  c  without  re-setting 
the  slide,  and  without  even  stopping  to  determine  the 


THE  SLIDE  RULE  61 

numerical  value  of  a/6.  Similarly  a?b  will  be  found 
on  A  opposite  b  on  the  B  scale  if  1  on  C  is  set  to  a 
on  D;  and  \/ab  will  be  found  on  D  under  b  on  B  if  1 
on  B  is  set  to  a  on  A.  A  series  of  fractions  like  a/m, 
b/m,  c/m,  d/m,  . .  .  can  be  read  off  by  merely  setting  the 
slide  rule  for  l/m  and  looking  opposite  a,  6,  c,  d,  ... 
Most  slide  rules  have  two  " cylinder  points"  marked 
off  on  the  C  scale  at  V  (4/ir)  and  V  (40/7r) ;  by  placing 
either  one  of  these  opposite  the  diameter  of  a  cylinder 
the  length  of  the  cylinder  on  B  will  indicate  its  volume 
on  A. 

Determination  of  Circular  Functions. — By  removing 
the  slide,  turning  it  over,  and  replacing  it  so  that  the 
ends  of  the  S  and  T  scales  coincide  with  the  ends  of 
the  A  and  D  scales  respectively,  the  sine  of  any  angle 
will  be  found  on  A  opposite  the  number  of  degrees  on 
S;  and  the  tangent  will  be  found  on  D  opposite  the 
number  of  degrees  on  T.  The  decimal  point  is  located 
by  recalling  the  facts  that  sin  90°  =  1  and  tan  45°  =  1. 

By  drawing  a  right-angled  triangle  the  student 
will  see  that  tan  (45  +  a)  is  the  reciprocal  of  tan 
(45  —  a),  and  this  fact  is  made  use  of  if  tangents  of 
the  larger  angles  are  to  be  obtained  from  the  slide  rule. 
For  sines  less  than  .01  and  tangents  less  than  .1  dif- 
ferent types  of  slide  rule  employ  different  methods, 
usually  based  upon  the  formulae  tan  d  =  sin  d  =5. 

The  scale  between  S  and  T  is  generally  marked  L 
and  is  used  for  obtaining  logarithms.  Set  1  on  C  oppo- 
site n  on  D  and  log  n  will  be  found  on  L  opposite  a 
special  mark  on  the  back  of  the  slide  rule. 


62 


GRAPHIC  REPRESENTATION 


VII.  GRAPHIC  REPRESENTATION 

The  position  of  any  point  on  the  surface  of  the  earth 
may  be  indicated  by  two  numbers :  one,  the  longitude 
of  the  point,  expresses  its  distance  to  the  east  or  west 
of  an  arbitrary  line,  the  meridian  of  reference;  the 
other,  its  latitude,  gives  its  distance  north  or  south 
of  another  line. 

In  almost  all  branches  of  science  the  opposite  t)f  this 
process  has  been  extensively  used,  and  the  location  of 
a  point  on  a  diagram  is  made  to  represent  the  numer- 
ical values  of  two  quantities  that  are  in  any  way  related 
to  each  other. 

Graphic  Diagrams. — If  two  straight  lines  are  drawn, 
one  horizontal  and  one  vertical,  the  position  of  any 
point  in  the  same  plane  can  be  represented  by  two 

numbers,  one  giving  its 
distance  to  the  right  of 
the  vertical  line,  the 
other  its  distance  above 
the  horizontal  line;  and 
conversely  any  two 
numbers  can  be  rep- 
resented by  properly 
locating  the  point. 

Paper  that  has  been 
ruled   in  small   squares 
is  convenient,  although 
not  necessary,  for  con- 
structing what  are  known  as  graphic  diagrams.    A  con- 
venient horizontal  line  is  chosen  for  one  axis  of  reference 
and  is  called  the  x-axis,  and  a  vertical  line  of  reference  is 


•X 


GRAPHIC  REPRESENTATION 


63 


called  the  i/-axis.  To  represent  the  numbers  5  and  3 
a  point  is  located  5  units  to  the  right  of  the  y-axis  and 
3  units  above  the  x-axis,  and  is  spoken  of  as  the  point 
(5,3).  The  above  diagram  shows  this  point  and  also 
the  points  (7,1)  and  (6,0)  by  small  dots;  and  by  going 
to  the  left  and  below  the  axes  it  is  possible  to  represent 
negative  as  well  as  positive  values,  as  is  seen  by  the 
small  crosses  located  at  (-f  6,  —  1)  (—  2,  +  3),  and 
(-  2,  -  1). 

The  following  table  contains  different  values  of  x 
and  the  corresponding  values  of  y.  Pick  out  the  values 
of  both  x  and  y  that  are  algebraically  largest  and 
smallest  and  mark  the  axes  of  reference  in  your  note- 
book in  such  a  position  that  there  will  be  space  enough 


x 

y 

1 

i 

2 

2 

1 

3 

2 

4 

3 

5 

4 

6 

5 

5 

(} 

4 

7 

3 

6 

2 

7 

1 

8 

0 

X 

y 

9  |  -  1 

10 

0 

11  1  -  1 

12 

_  2 

13 

-  3 

12 

-  4 

11 

-  5 

10 

-  6 

9 

-  5 

8 

-  6 

7 

v        o 
1  )       " 

6 

-  8,  -  10 

x 

V 

1 

-7,  -9 

-1 

-  1,  -5 

2 

-  10 

5 

-  11 

2 

-  8 

3 

-  11 

-2 

0 

-3 

-  1,  -5 

-4 

-  2,  -  4 

0 

0,  -6 

-5 

-  3 

-2 

-  6 

for  the  points  representing  the  extreme  values.  Then 
take  each  pair  of  values  in  succession  and  plot  each 
corresponding  point. 

Graphic  diagrams  are  used  chiefly  to  show  the  rela- 
tion ^between  two  variable  quantities.  A  quantity  that 
is  varied  arbitrarily  is  usually  represented  by  x,  or  hori- 


64  GRAPHIC  REPRESENTATION 

zontally,  while  the  dependent  variable  or  related  effect 
is  represented  by  y  or  vertically.  Thus,  if  a  gas  is  com- 
pressed it  will  have  a  certain  volume  for  each  amount 
of  pressure,  and  its  condition  at  any  time  can  be 
represented  by  plotting  vertically  the  resultant  volume 
due  to  the  independently  varied  pressure  that  is  repre- 
sented horizontally. 

The  following  table  is  an  example  of  the  typical 
fluctuations  in  body-temperature  of  a  normal  indi- 
vidual. Plot  the  corresponding  points  on  a  graphic 
diagram  in  which  each  small  square  represents  one 
hour  horizontally  and  one-tenth  of  a 


hour 


12 
1 
2 
3 
4 
5 
6 


9 

10 
11 
12 


A.M.          P.M. 


temP-  degree  vertically.  In  order  to  save 
space  allow  only  enough  vertical 
distance  for  the  plotted  points, 

36.8  37.4  namely  13  squares,  and  lay  off  a 
numbered  scale  on  the  ?/-axis  to 

36^4    37.5     show    that    the    z-axis  or    zero    of 

?«'*    07 '«     temperature  must  be  considered  as 
36.6     37.6 


36.9    37.4 


a         c 
36.6     37.5 


situated    far    below    the     diagram. 
QA'O    QV'^     Note    how   much    more    " graphic" 


36.9    37.6 


37 .1  37  A  the  diagram  is  than  the  table;  how 
37  •  2  37 . 4  ft  shows  at  a  glance  facts  that  could 
37^4  36^9  be  gleaned  from  the  table  only  with 


much  greater  effort. 


Make  another  diagram  which  is  an  exact  duplicate 
of  this  one.  In  one  diagram  draw  a  straight  line  from 
each  point  to  the  next  one  so  that  all  the  points  are 
united  by  a  broken  line.  In  the  other  try  to  draw  a 
smooth  curve  which  shall  follow  the  general  trend  of 
the  points  as  nearly  as  possible  and  lie  above  as  many 
points,  approximately,  as  it  lies  below.  It  should 


GRAPHIC  REPRESENTATION  65 

show  not  more  than  one  downward  swing  and  one 
larger  upward  swing. 

The  method  of  connecting  points  with  a  broken 
line  is  used  when  it  is  desired  to  represent  the  sepa- 
rate tabular  values  without  implying  anything  about 
intermediate  values,  as  in  most  cases  where  the  law 
of  variation  is  unknown.  The  method  of  drawing 
an  approximate  curve  is  known  as  " smoothing"  the 
graphic  diagram ;  it  is  used  only  when  the  discrepancies 
between  the  actual  points  and  the  smoothed  curve  are 
no  greater  than  can  be  accounted  for  by  unavoidable 
errors  in  obtaining  the  numerical  values,  or  when  they 
are  known  to  arise  from  unimportant  causes.  A 
third  method  is  possible  when  the  quantities  vary 
according  to  some  mathematical  law.  In  such  case  a 
smooth  curve  can  generally  be  so  drawn  as  to  pass 
exactly  through  all  of  the  plotted  points,  as  will  be 
seen  in  the  examples  given  later;  and  if  an  equation 
is  to  be  represented  graphically  only  enough  points 
need  be  plotted  to  allow  the  shape  of  such  a  curve  to 
be  satisfactorily  determined. 

Graph  of  an  Equation. — A  single  equation  in  two  un- 
known quantities  has  an  infinite  number  of  solutions. 
Any  value  may  be  assigned  to  one  unknown 
quantity  and  the  corresponding  value  cal-  a 
culated  for  the  other.  A  single  solution  0 
can  be  represented  by  a  point  on  a  graphic  1 


diagram   and   the  totality  of  such   points 
form   a   curved  or   straight    line,  which  is        4 
called  the  " curve"  of  the  equation.     Thus 
if  y  =  2x  +  3  the  values  of  x  and  y  given     —3 
in    the    table    are    all  solutions   and    may 
5 


-1 
-2 


-4 


y 


3 

5 

7 

9 

11 

1 

-1 
-3 
-5 


66  GRAPHIC  REPRESENTATION 

all  be  plotted  on  a  graphic  diagram.  The  " curve"  of 
this  equation  is  easily  seen  to  be  a  sloping  straight  line, 
and  a  trial  will  show  that  fractional  values  of  x  or  y 
will  also  give  points  that  lie  exactly  on  the  line. 

In  plotting  an  equation  the  table  of  values  should 
always  be  first  calculated,  substituting  successive  posi- 
tive and  negative  integral  values  of  x  (and  fractional 
values  if  necessary)  and  solving  for  y,  and  then  the 
available  space  on  the  paper  considered  and  the  axes 
drawn  in  a  suitable  location.  If  the  ^-values,  or 
" abscissae"  as  they  are  called,  are  either  very  large  or 
very  small  in  comparison  with  the  ?/-values  or  "ordi- 
nates"  the  two  quantities  may  be  plotted  with  scales 
of  different  size;  but  along  each  axis,  considered  by 
itself,  equal  distances  must  always  correspond  to  equal 
numerical  differences,  and  x-values  must  always 
increase  to  the  right  and  y- values  increase  upward.  It 
is  often  advisable,  in  cases  where  the  curve  will  have  a 
fairly  uniform  slope  and  the  relative  values  of  the  x-unii 
and  the  i/-unit  are  unimportant,  as  in  the  extrapolation 
diagram,  page  77,  and  in  the  greater  part  of  the  diagram 
on  page  132,  to  choose  such  scales  that  this  slope  will  be 
approximately  45°.  In  the  following  exercises  of  this 
lesson,  however,  a  single  square  of  the  paper  is  to  repre- 
sent one  unit,  in  each  direction,  except  where  otherwise 
specified. 

On  the  squared  paper  of  the  note-book  lay  off  a 
rectangle  10  squares  wide  and  20  squares  high.  Draw 
the  axes  so  the  rectangle  extends  from  x  =  —5  to 
x  =  +5  and  from  y  =  -10  to  y  =  +10.  Within 
the  limits  of  this  rectangle  plot  the  curve  of  the  equa- 
tion y  —  2x  by  assigning  integral  values  to  x  from 


GRAPHIC  REPRESENTATION  67 

—  5  to  +5,  and  discarding  any  points  whose  y  values 
do  not  come  within  ±10.  On  the  same  diagram  plot 
y  =  ^x}  y  =  —  %x,  and  y  =  2x  +  4.  How  many  of  these 
are  straight  lines?  On  another  diagram,  between  the 
limits  y  =  —  5  and+  25,  and  x  =  =«=  10,  plot  the  following : 

y  =  x2',  y  =  x2/2;  y  =  x2/W',  y  =  -x2/lQ;  y  =  4; 
x  =  3  (hint:  substitute  various  values  of  y  and  cal- 
culate the  corresponding  values  of  x). 

All  curves  of  the  form  y  =  ax2  are  similar  figures; 
y  =x2  and  y  =  -fax2  differ  in  size  but  the  portion  of 
the  latter  extending  from  x  =  —  5  to  x  =  +5  is  of 
exactly  the  same  shape  as  the  part  of  the  former  that  is 
included  between  x  =  —  1  and  x  =  + 1 .  The  curve  is 
called  a  parabola. 

Plot  y  =  x*  using  the  scales  that  you  think  best. 
Carry  the  ^-values  only  far  enough  to  make  the  y- 
values  increase  or  decrease  rapidly. 

Plot  y  =  30/x  with  enough  points  to  show  clearly 
the  form  of  both  branches  of  the  curve;  plot  on  the 
same  diagram  y  =  l/x. 

Draw  the  curve  of  y  =  l/x2.  How  would  the  curve 
of  y  =  30/x2  differ  from  it.  How  would  y  =  —l/x2 
compare  with  it? 

Plot  y  =  log  x  from  x  =  0  to  x  =  10  using  a  large 
scale  on  the  z-axis  (for  example,  five  squares  for  each 
unit),  and  one  about  twice  as  large  on  the  y-axis. 
Locate  as  many  points  as  are  needed  to  determine 
a  smooth  curve;  the  whole  numbers  from  1  to  10 
ought  to  be  sufficient. 

T«rn  back  to  the  page  left  vacant  in  the  lesson  on 
logarithms,  and  plot  carefully  the  equation  y  =  e~x2 
from  x  =  —0.5  to  x  -  2.5  or  3.  Turn  the  note-book 


68  GRAPHIC  ANALYSIS 

so  that  the  z-axis  will  lie  lengthwise  of  the  page  and 
use  as  large  a  scale  as  possible  (say  20  squares  =  1.0). 
If  y  equals  +0.7  when  x  =  +0.6  what  does  y  equal 
when  x  =  —0.6?  Note  what  value  of  x  is  required  in 
order  to  make  the  corresponding  ordinate  indistinguish- 
able from  zero  on  your  diagram.  The  importance  of 
this  curve  will  be  seen  later  when  taking  up  the  subject 
of  errors  of  observation. 


VIII.  GRAPHIC  ANALYSIS 

Apparatus. — Fine  black  silk  thread;    slide  rule. 
The  line  plotted  on  a  previous  diagram  for  y  = 
2x  +  4  will  be  seen  to  have  each  of  its  points  four  units 
higher  than  a  .corresponding  point  on  the  line  y  =  2x ; 
in  fact  this  could  have  been  inferred 
without  plotting  either  of  the  lines, 
f/  for  it  is  evident  that  whatever  may 

be  the  value  of  x  the  y  of  the  former 
equation  is  greater  than  that  of  the 
y=4  latter  by  four  units. 

";B~  In    connection    with  the    study 

-l/c^  of    physical    changes    it   is   more 

convenient  to  put  such  an  equation 
in  the  form  y  =  4  +  2x   and  to 


0 


X      consider  the  ^/-values  of  the  curve 


A  as  the  sum  of  the  constant  4  (AB 

in  the  diagram)  and  2x  (AC  in  the 
diagram).  BD  is  made  equal  to  AC  so  that  the  new 
?/-value,  or  AD,  is  AB  +  AC,  or  4  +  2x. 

The  plot  of  y  =  a  +  bx. — Accordingly,  to  plot  such 
a  curve  as  y  =  a  +  bx,  whatever  values  a  and  b  may 


GRAPHIC  ANALYSIS  69 

have,  it  is  necessary  only  to  mark  a  new  zero-point,  or 
"origin"  of  the  graphic  diagram,  as  it  is  called,  at  a 
distance  a  above  the  old  origin,  and  to  plot  y  =  bx  from 
this  new  starting-point.  Notice  that  b  is  the  numerical 
tangent  of  the  slope  of  the  line,  and  a  is  the  "intercept" 
on  the  ?/-axis,  or  the  distance  above  O  to  the  point 
where  the  ?/-axis  is  cut  by  the  curve. 

Draw  the  following  equations  on  one  diagram  by 
laying  off  the  ^-intercept  (OO'  in  the  diagram)  and 
drawing  a  straight  line  that  has  the  proper  slope 
(ratio  of  BD  to  O'B  in  the  diagram) :  y  =  2  +  3x; 
y  =  1  +  0.5z;  y  =  -  2  +  0.5z;  y  =  2  +  0.5z; 
y  =  2  +0.25:r;  y  =  2  -  0.25z;  y  =  2  -  0.5z. 

If  there  is  any  doubt  as  to  the  meaning  of  a  nega- 
tive slope  a  few  points  should  be  plotted  from  the 
equation  in  the  usual  manner. 

It  should  be  noticed  that  the  form  y  =  a  +  bx 
cannot  be  used  for  every  straight  line  that  can  be 
drawn.  The  general  equation  of  the  straight  line  is 
Ax  +  By  -f  C  =  0.  This  can  be  made  to  represent 
lines  parallel  to  the  i/-axis  by  putting  B  equal  to  zero; 
such  lines,  however,  are  never  needed  for  expressing 
physical  changes. 

If  two  quantities  are  always  proportional  or  vary 
directly  with  one  another,  as  in  the  case  of  the  absolute 
temperature  of  a  gas  and  its  resultant  pressure,  what 
kind  of  a  graph  would  be  obtained  by  plotting  one 
of  them  as  x  and  the  other  as  yl 

If  equal  changes  in  one  quantity  always  correspond 
to  -equal  changes  in  another  what  kind  of  a  graph 
expresses  their  relationship?  Explain  just  what  the 
difference  is  between  this  graph  and  the  previous  one. 


70  GRAPHIC  ANALYSIS 

An  example  of  this  kind  of  relationship  is  given  by  a 
metal  bar  which  has  a  certain  length  at  a  certain 
temperature  and  undergoes  changes  in  length  which 
are  accurately  proportional  to  the  changes  in  its  tem- 
perature. If  there  is  any  trouble  in  answering  the  ques- 
tion an  arbitrary  point,  say  the  point,  (a,  6),  should  be 
plotted,  and  then  two  or  three  other  points,  (xi,  y\), 
etc.,  which  are  so  situated  that  (xi  —  a)/(yi  —  b)  = 
(x2  -  a)/(yz  -  b)  =  (x3  -  a)/(y3  -  b),  etc. 

The  "Black  Thread"  Method.— If  a  set  of  experi- 
mental measurements,  such  as  those  of  the  temperature 
and  length  of  a  metal  bar,  are  found  to  correspond 
approximately  to  the  " straight  line  law"  they  may 
be  plotted  and  "  smoothed ''  by  drawing  the  straight 
line  that  comes  closest  to  all  the  points.  This  is  called 
the  " black  thread"  method  because  the  position  of  the 
line  is  decided  upon  with  the  aid  of  a  stretched  thread 

instead  of  a  ruler;  the  thread  and  the  points 
y       can  all  be  seen  at  the  same  time,  while  a  ruler 

would  hide  half  of  the  points  if  it  was   so 


1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 


9  8 

g'5  placed  as  to  lie  evenly  among  them. 

8-0  Plot  these  tabular  values  as  accurately  as 

6  '7  possible,  marking   each   point    by  a  minute 

6-5  dot  surrounded  by  a  small    circle,   or  by  a 

5  ]  5  cross  composed  of  a  short  vertical  line  to  mark 

5-0  the  exact  value  of  the  abscissa  and  a  short 

3  '  9  horizontal  line  at  the  exact  height  of  the  ordi- 


2.3 


nate. 

See  that  the  page  of  the  note-book  rests  in 

a  perfectly  flat  position  and  stretch  a  fine  black 
silk  thread  in  a  straight  line  that  follows  the  general  direc- 
tion of  the  points.  Move  it  a  trifle  upward  and  downward 


GRAPHIC  ANALYSIS  71 

along  the  ?/-axis,  also  rotate  it  slightly,  both  clockwise 
and  counter-clockwise.  When  you  finally  get  the  posi- 
tion that  you  think  lies  most  closely  and  evenly  among 
the  points  notice  exactly  where  the  thread  cuts  both 
the  x-axis  and  the  i/-axis,  and  from  these  two  values 
calculate  its  slope,  noticing  whether  the  value  is  posi- 
tive or  negative.  What  is  the  equation  of  the  line  indi- 
cated by  the  thread,  expressed  in  the  form  y  =  a  +  bxl 

What  are  the  distances  intercepted  on  the  two  axes 
by  the  line  x/a  +  y/b  =;  1?  How  can  you  use  this 
form  to  make  it  easier  to  calculate  the  equation  of  a 
black  thread  determination? 

The  Plot  of  y  =  a  +  bx  +  ex2. — Just  as  the  curve 
y  =  a  +  bx  can  be  considered  as  having  its  ordinate 
for  each  value  of  x  built  up  of  the  ordinate  a  plus  the 
ordinate  bx,  so  the  more  general  form  y  =  a  -\-  bx  -\-  ex2 
can  be  considered  as  made  up  of  the  straight  line 
y  =  a  +  bx,  on  which  is  superimposed  the  parabola 
y  =  ex2.  Curiously  enough,  this  also  represents  a 
parabola  in  every  case  where  c  is  different  from  zero. 

On  a  single  graphic  diagram  plot  y  =  —  O.lx2  and 
y  =  3  +  0.5x.  Then  add  the  ordinates  of  the  former 
algebraically  to  the  latter  and  draw  as  smoothly  as 
possible  the  resultant  curve  y  =  3  +  0.5z  —  O.lz2. 

If  an  experimental  curve  looks  as  if  it  could  be 
represented  by  y  =  a  +  bx  +  ex2  it  is  possible  to 
draw  an  approximate  tangent,  find  its  equation,  and 
then  determine  the  coefficient  of  x2,  but  it  is  usually 
more  satisfactory  to  proceed  as  in  the  following  example, 
completing  a  free-hand  parabola  as  far  as  the  point 
where  it  runs  horizontally  if  such  a  point  is  not  already 
present. 


72 


GRAPHIC  ANALYSIS 


The  density  of  water  at  different  temperatures  is 
given  in  the  table.  If  the  density  is  called  y  and  the 
temperature  x  what  are  the  numerical  values  of  the 
coefficients  in  the  equation  y  =  a  +  bx  -\-  c#2? 


temp. 

dens. 

0 
2 
4 

6 
8 
10 

.99984 
.99994 
.99997 
.99994 
.  99984 
.99970 

12 
14 
16 

18 
20 

.  99950 
.99924 
.99894 
.99859 
.99820 

22 
24 
26 
28 
30 

.99777 
.99729 
.99678 
.99623 
.99564 

2    3 

13 

6 

27 

8 

47 

10 

73 

12 

103 

14 

138 

16 

177 

18 

220 

20 

268 

22 

319 

24 

374 

26 

433 

The  first  step  is  to  make  a  careful  graph.  The 
scale  on  the  7/-axis  needs  to  extend  only  through  the 
numbers  .995,  .996,  .997,  .998,  .999,  1.000.  It  is  appar- 
ent that  the  curve  is  very  much  like  a  parabola,  and  if 
we  should  arbitrarily  put  the  origin  of  the  graphic 
diagram  at  the  point  (4.,  .99997)  the  equation  would 
be  y  =  —  mx*.  The  positive  values  of  its  negative 
ordinates  will  be  obtained  by  subtracting  each  tabular 
density  from  .99997,  and  the  new  values  of  x  will  be 
4  less  than  before.  The  results  are  shown  in  the  next 
table,  with  their  decimal  points  omitted.  Now,  with 


GRAPHIC  ANALYSIS  73 

your  slide  rule,  read  off  the  square  roots  of  the  num- 
bers in  column  y  and  enter  them  in  the  column  ^y. 
If  y  is  proportional  to  x2,  as  shown  by  the  equation 
y  =  —  mx2,  then  the  square  root  of  y  must  be  pro- 
portional to  x  itself,  and  their  relative  magnitudes  can 
be  determined  by  the  black  thread  method. 

Plot  the  corresponding  values  of  x  and  ^y  and 
determine  the  slope  with  the  black  thread,  remem- 
bering that  the  line  must  pass  through  the  origin. 
Your  value  of  the  tangent  should  not  differ  greatly 
from  f  or  0.83. 

From  V  y  =  .83x  it  follows  that  y  =  —  .69z2,  y  being 
expressed  in  hundred-thousandths,  and  with  the  aid 
of  the  slide  rule  the  values  of  .69a:2  are  read  off  as  shown 
in  the  table.  In  the  last  column  they  have  been  in- 
creased by  3  to  make  it  more  convenient  to  plot  them. 

On  the  same  graphic  diagram  plot 
the  values  of  y  +  3  in  .OOOOl's  down-      x  l-QQx*  ^  +  3 


10 


20 
22 


44 
69 
99 


14 

28 

47 

72 

102 


ward  from  the  line  y  =   1.00000,  re-  o         0   ; 
membering  that  x  =  0  is  now  at  the 
apex   of  the  curve,   and  notice  how  6  |     25 
closely  the  graph  of  the  equation  coin- 
cides with  the   graph  of  the  experi-  12 
mental  values.  J^      |35    I    138 
To    reduce  the   equation    100000?/  18  \   224   |   227 
=  —  .69x2  to  the  original  axes  notice 
that   any   point  whose  abscissa  and  24 
ordinate  are  (x,  y)  with  reference  to  26 


276  279 

334  |    337 

398  I    401 

467  470 


the  new  axes  will  have  an  abscissa 
of  *  +  4   (call  it  x')   and  an  ordinate  of  y  -j-  .99997 
( =  y'}  when  referred  to  the  old  axes.     Hence  x  =  x' 
—  4  and  y  =  yf  —  .99997;  substituting  these  values  in 


74  GRAPHIC  ANALYSIS 

100000?/  =    -.69z2  we  obtain  100000  (yf  -  .99997)  = 
-  69  (xf  —  4)2,  or,  dropping  the  accents  and  simplifying: 

y  =  .99986  +  .0000552  re  -  .0000069  a;2. 

Writing  d  for  the  density  of  water  and  t  for  centi- 
grade temperature  this  becomes 

d  =  .99986  +  .0000552  t  -  .0000069  t\ 

Linear  Relationship  by  Change  of  Variables. — Any 
law  of  change  can  be  expressed  by  the  formula 
y  =  a  +  bx  +  ex2  +  dx*  +  . .  if  enough  terms  are  used, 
but  the  method  soon  becomes  difficult  to  handle. 
Sometimes  the  appearance  of  the  curve  makes  it  pos- 
sible to  guess  that  its  equation  is  of  some  particular 
form,  such  as 

ax 

y  =  a/x,  or  y  =  e<™,  or  y  =  &  _jTT, 

or  the  form  may  be  known  from  theory.  In  such  cases 
it  is  best  to  make  use  of  a  procedure  similar  to  that 
employed  for  determining  m  in  the  equation  above, 

y  =  —  mx2.     For  example  y  =  ,       -  will  give  a  curved 

0  ~~r~  30 

line,  but  by  calculating  y/x  it  will  be  found  that  for 
this  particular  equation  the  graph  of  y  and  y/x  will  be 
a  straight  line. 

The  volume  of  a  certain  mass  of  air  was  found  to 
vary,  with  the  pressure  to  which  it  was  subjected, 
according  to  the  numbers  in  the  following  table. 
Represent  the  relation  between  volume  and  pressure 
by  an  equation. 


4 

3 

3 

13 

19 


GRAPHIC  ANALYSIS  75 

Here  it  is  known  from  Boyle's  law  that  the  pressure 
and  volume  are  inversely  proportional,  or  v  =  a/p.    If 
the   pressure  is  denoted  by  x   and   the 
volume   by  y  a   curve  can  be   plotted,     /^ .    ,  K,x 
but  a  straight  line  ought  to  be  obtained 
by  plotting  y  and  1/x.  j  5 

Choose  the  x-  and  ^/-scales  so  that  the        2  5 

line  is  neither  very  steep  nor  very  flat, 
use  the  black  thread  method,  and  deter-        3.5 
mine  the  value  of  a  in  the  above  equa-          -^ 
tion. 

Graphic  Interpolation. — In  the  case  of  most  experi- 
mental curves  and  mathematical  equations  the  fact 
that  the  positions  of  certain  points  are  known  makes 
it  possible  to  draw  a  smooth  curve  through  them  and 
thus  determine  the  location  of  any  intermediate  point. 

Turn  back  to  the  graph  of  y  =  log  x.  How  high  is 
the  curve  above  the  base-line  for  x  =  3.5?  Do  not 
draw  the  ordinate  but  measure  it  accurately.  In  the 
same  way  measure  the  logarithms  of  2.5,  1.2,  and  0.8. 
When  the  independent  variable  in  this  equation  is 
less  than  1  the  ordinate  should  be  measured  downward 
past  the  curve  to  the  next  (negative)  whole  number, 
and  then  another  measurement  made  from  this  point 
upward  to  the  curve.  This  will  put  a  number  like 
—  3.8  in  the  form  of  minus  four  plus  the  other  two- 
tenths,  or  4.2,  as  a  logarithm  is  usually  written. 

Join  the  two  points  of  the  curve  (2,  log  2)  and  (3, 
log  3)  by  a  straight  line  on  your  diagram.  In  the  lesson 
on* logarithms  it  was  assumed  that  the  value  of  log 
3142  was  one-fifth  of  the  way  from  log  3140  to  log  3152 
because  the  natural  number  3142  was  one-fifth  of  the 


76  GRAPHIC  ANALYSIS 

way  from  3140  to  3150.  If  this  assumption  were 
absolutely  accurate  what  would  be  the  shape  of  the 
logarithmic  curve  between  x  =  3140  and  x  =  3150? 
Can  you  make  the  same  assumption  for  the  stretch 
of  curve  lying  between  (2,  .301)  and  (3,  .477)?  Com- 
pare the  ordinate  of  the  middle  point  of  the  chord  just 
drawn  with  the  interpolated  value  of  log  2.5  and  with 
the  value  of  log  2.5  obtained  from  the  tables.  Why  is 
the  first  assumption  justifiable  in  using  a  table  of 
logarithms? 

Turn  to  the  5-place  table  on  page  144  and  notice 
that  the  slope  of  y  =  log  x  varies  so  rapidly  between 
x  =  1.000  and  x  =  1.100  that  the  "argument"  (or 
natural  number,  or  antilogarithm)  has  to  be  given  at 
intervals  of  .001  in  order  to  make  the  process  of  inter- 
polation trustworthy  to  5  decimal  places.  Beyond 
x  =  1.100,  however,  intervals  as  large  as  .01  can  be 
used  without  the  chord  deviating  from  the  curve 
enough  to  affect  the  fifth  decimal  place. 

What  logarithm  is  accurately  represented  by  the 
ordinate  of  the  middle  of  your  chord,  and  which  of 
the  formulae  for  approximate  calculation  with  small 
magnitudes  is  illustrated  by  this  diagram? 

Plot  the  density  of  water  on  a  large  scale,  allowing 
the  height  of  a  small  square  to  represent  .00001  and 
using  five  squares  horizontally  for  2  degrees.  Mark 
down  the  points  0°,  2°,  4°,  6°,  8°,  and  draw  a  smooth 
free-hand  curve  through  them;  unite  them  also  with 
chords,  and  find  the  density  at  1  °  and  3  °  or  at  5  °  and 
7°  by  both  methods  of  interpolation.  For  which  points 
is  the  arithmetical  method  liable  to  be  erroneous  and 
for  what  values  could  it  probably  be  trusted? 


THE  PRINCIPLE  OF  COINCIDENCE 


77 


Graphic  interpolation  is  especially  useful  in  cases 
of  smoothed  curves.  If  two  successive  ordinates  hap- 
pened to  have  errors  of  the  same  sign  the  value  obtained 
by  arithmetical  interpolation  would  be  less  accurate 
than  that  obtained  graphically. 

From  your  graphic  diagram  (see  page  64)  find  the 
most  probable  value  of  the  average  temperature  of  a 
healthy  individual  at  9:30  A.M. 

Extrapolation. — The  process  of  finding  the  value  of 
a  function  beyond  its  known  values  instead  of  between 
them  is  called  extrapolation.    It  needs 
to  be  employed  with  extreme  caution 
where   the    general  law  of   change    is 
unknown. 

Draw  a  graphic  diagram  of  the 
population  of  California  at  ten-year 
intervals  as  given  in  the  table,  con- 
necting the  points  with  a  smooth 
curve.  Continue  the  curve  so  as  to 
determine  the  value  for  1910. 

Reconstruct  the  curve,  if  necessary,  so  as  to  give 
2.38  X  106  for  1910,  and  find  what  the  population  will 
be  in  1920. 


year. 

pop'l'n. 

1850 

93,000 

1860 

380,000 

1870 

560,000 

1880 

865,000 

1890 

1208,000 

1900 

1485,000 

1910 

IX.   THE  PRINCIPLE  OF  COINCIDENCE 

Apparatus. — Two  metre  sticks;  card-board;  knife; 
square  wooden  block;  slide  rule. 

Measure  one  inch  in  centimetres,  millimetres,  and 
estimated  tenths  of  a  millimetre.  Measure  a  length 
of  two  inches  in  the  same  way,  and  divide  the  result 
by  two.  Take  one-tenth  of  the  measured  length  of 


78 


THE  PRINCIPLE  OF  COINCIDENCE 


ten  inches  and  compare  with  the  two  other  results. 
Notice  that  a  large  quantity  can  usually  be  measured 
more  accurately  than  a  small  one. 

Measurement  by  Estimation. — In  order  to  make  a 
more  accurate  determination  of  the  length  of  an  inch 
lay  one  metre  stick  on  the  table  with  the  metric  gradu- 
ations upward  and  place  the  other  beside  it  with  the 
graduation  in  inches  and  tenths  of  an  inch  upward. 
See  that  the  scales  are  in  close  contact  and  note  down 
the  number  of  centimetres  and  hundredths  indicated 
on  one  scale  by  the  mark  1  inch  on  the  other.  Note 
also  the  indication  of  the  mark  31  inches.  Move  one 
scale  along  the  other  at  random  and  repeat  the  obser- 
vations until  ten  sets  have  been  taken.  The  distance 
measured  should  always  be  the  same,  30  inches,  but 
it  should  be  measured  at  other  places,  such  as  the 
mark  4  and  the  mark  34  Never  use  the  mark  0  at 
the  very  end  of  the  stick,  as  it  is  often  inaccurate  on 
account  of  wear. 

Tabulate  the  results  in  a  form  like  the  following: 


1st  line 

2dline 

Interval 

Cm  scale 
Inch  scale 

20.05 
1.00 

45.50 
11.00 

25.45 

10.00 

Cm 
Inch 

50.00 
2.00 

75.47 
12.00 

25.47 

10.00 

Cm 
Inch 

10.06 
1.50 

35.46 
10.50 

25.40 

10.00 

Sum     .... 
Average 

254.02 
25.402 

100.00 
10.000 

Length  of  an  inch:  —  2.5402  cm. 


THE  PRINCIPLE  OF  COINCIDENCE 


79 


Measurement  by  Coincidence. — Set  the  two  scales 
so  that  any  whole  number  of  inches  near  one  end  of 
one  metre  stick  is  exactly  opposite  some  whole  num- 
ber of  centimetres  on  the  other.  Hold  the  two  sticks 
firmly  together  and  look  again  to  see  that  the  coin- 
cidence is  exact.  Find  another  place,  at  least  30  cm 
distant  from  this  point  and  preferably  further,  where 
there  is  another  exact  coincidence,  this  time  between 
any  centimetre  or  millimetre  graduation  and  any 
graduation  of  inches  or  tenths.  Try  to  decide  whether 
the  coincidence  is  exact  or  whether  the  imaginary 
central  axis  of  one  line  lies  a  little  beyond  the  other, 
and  if  the  coincidence  is  faulty  choose  a  better  one 
elsewhere. 

Record  the  results  as  in  the  specimen  table,  re-set 
the  two  scales,  and  repeat  until  10  determinations  have 
been  made. 


1st  line 

2d  line 

interval 

cm  in  1  in. 

Cm 
In. 

90.000 
4.000 

35.400 
25.500 

54.600 
21.500 

2.5395 

Cm 
In. 

15.000 
6.000 

71.400 
28.200 

56.400 
22.200 

2.5405 

Average  . 

2.5401 

It  is  not  absolutely  accurate  to  assume  that  the 
coincidences  are  exact  to  a  hundredth  of  the  smallest 
graduation  of  the  scale,  but  they  certainly  ought  never 
to  have  an  error  of  as  much  as  half  a  tenth,  or  .05, 
and^arely  as  much  as  .01  if  the  work  is  carefully  done. 

The  Vernier. — Rule  a  straight  line  along  a  strip  of 
cardboard  that  measures  about  10  X  25  cm.  Lay  off 


80  THE  PRINCIPLE  OF  COINCIDENCE 

a  scale  of  centimetres  below  it  and  a  scale  of  ten  mul- 
tiples of  9  mm.  above  it  as  shown  in  the  diagram.  Cut 
the  cardboard  from  A  to  B  and  from  B  to  C,  so  that 
the  short  scale  can  slide  along  the  centimetre  scale. 
The  small  scale  is  called  a  vernier  and  is  used  to  indi- 
cate tenths  of  the  divisions  on  the  large  scale.  Move 
it  to  the  right  very  slowly,  making  its  first  division 
coincide  with  a  division  on  the  main  scale,  then  further 
until  the  second  division  does  likewise,  and  so  on, 
until  the  tenth  vernier  division  is  opposite  some 
division  of  the  other  scale.  Notice  that  the  zero  of 
the  vernier  has  been  moved  just  one  centimetre  during 
the  process;  hence  if  some  other  division  of  the  vernier, 
such  as  No.  3,  coincides  with  any  centimetre  division 
the  zero  of  the  vernier  must  be  three-tenths  of  an  inter- 
val beyond  some  scale  mark. 
A 


i.     ;TT      .io 

i    i   i   i    1  i    i   i   i   1 

A 

B    1    I    '    >    '    |    "    I    '    I    1    II    II 
\            lo                      5                       f|0                      1 
\^_           SCaU 

M    M    ]    1 
5                    2|o 

Separate  the  jaws  of  this  model  vernier  caliper  by  a 
distance  of  three  and  a  half  centimetres,  as  nearly  as 
you  can  by  estimation;  then  look  at  the  vernier  and 
notice  that  its  zero  is  beyond  the  mark  3  of  the  centi- 
metre scale  and  that  the  division  5  of  the  vernier  coin- 
cides with  some  division  (no  matter  what)  of  the  main 
scale.  The  reading  is  accordingly  3.5  cm.  In  the  same 
way  read  the  indicated  length  when  the  caliper  is  set 
at  random. 


THE  PRINCIPLE  OF^  COINCIDENCE  81 

Hold  the  two  parts  of  the  caliper  in  alignment  by 
pinching  the  line  BC  between  the  fore-finger  and  thumb 
of  both  hands  and  use  the  apparatus  to  make  twelve 
measurements  of  the  thickness  of  the  wooden  block  at 
different  positions  around  its  edge.  Find  the  average 
thickness  of  the  block,  carrying  the  calculation  out  to 
one  more  decimal  place  than  the  individual  measure- 
ments. 

Slide-Rule  Ratios. — Set  the  slide  rule  so  that  the 
number  seven  on  the  B  scale  is  under  22  on  A,  and  see 
if  the  end  of  B  comes  opposite  the  special  mark  TT  on 
the  A  scale.  If  not,  set  1  on  B  accurately  under  TT  on 
A  and  look  along  the  scales  until  two  graduations  are 
found  which  are  precisely  opposite.  See  if  these  two 
numbers  are  multiples  of  22  and  70,  and  if  so  try  to 
find  another  ratio  which  is  indistinguishable  from  TT 
and  is  not  identical  with  22/7. 

Numbers  like  the  22  and  66  on  the  back  of  the  slide 
rule  for  the  ratio  of  inches  and  centimetres  are  so 
chosen  as  to  give  the  correct  value  to  at  least  as  many 
significant  figures  as  can  be  read  with  the  apparatus 
used.  Some  slide  rules  give  the  equivalent  as  50  in.  = 
127  cm.  This  has  the  disadvantage  of  not  being  quite 
as  easy  to  set  on  the  A  and  B  scales  as  on  C  and  D, 
but  for  use  with  a  very  finely  graduated  instrument 
or  with  one  on  which  the  scales  are  20  inches  long 
instead  of  10  it  has  the  advantage  of  greater  accuracy, 
for  66.0000  •*•  26.0000  =  25.3846  while  127.0000  -5- 
50.00000  =  25.40000.  The  former  ratio  is  correct  to 
thr^e  significant  figures  only,  while  the  latter  is  correct 
to  five. 


82  MEASUREMENTS  AND  ERRORS 

The  number  of  avoirdupois  or  troy  grains  in  one 
gram  is  15.432.  Find  a  slide-rule  ratio  which  gives  this 
value  correct  to  four  significant  figure's. 

X.   MEASUREMENTS  AND  ERRORS 

Apparatus. — A  vernier  caliper  reading  to  tenths  of  a 
millimetre;  100  variates  (such  as  the  seeds  of  Phaseolus) 
for  measurement. 

Direct  and  Indirect  Measurements. — Measurements 
are  classified  as  direct  and  indirect.  The  ordinary  pro- 
cesses of  measuring  length  or  weight  are  direct,  because 
the  unknown  length  is  placed  beside  a  standard  series 
of  multiples  and  fractions  of  the  unit  of  length  and  is 
directly  compared  with  it,  and  an  unknown  weight 
is  directly  balanced  against  a  series  of  known  weights 
until  its  exact  equivalent  is  determined.  An  example  of 
an  indirect  measurement  is  the  usual  method  of  deter- 
mining density.  The  volume  of  an  object  and  its  mass 
are  determined  directly,  and  its  density  is  then  obtained 
from  the  direct  measurements  by  calculating  the  ratio 
of  mass  to  volume. 

Independent,  Dependent,  and  Conditioned  Measure- 
ments.— Measurements  are  also  classified  as  independ- 
ent, dependent,  and  conditioned. 

If  there  is  a  theoretical  relationship  that  must  always 
hold  good  between  different  quantities,  their  measure- 
ments are  said  to  be  conditioned.  An  example  of  this 
is  the  measurement  of  the  three  angles  of  any  triangle. 
Their  sum  must  necessarily  be  180°,  or  ir. 

Measurements  are  said  to  be  dependent  if  one 
measurement  is  allowed  to  influence  or  bias  the  ob- 


MEASUREMENTS  AND  ERRORS  83 

server  when  making  a  later  measurement  of  the  same 
quantity.  With  delicate  measurements  this  effect  is 
so  hard  to  avoid,  even  if  the  observer  has  the  best  of 
intentions,  that  it  is  always  advisable  to  guard  against 
it  by  some  such  device  as  hiding  the  scale  of  an  appar- 
atus from  view  until  after  the  indicating  mark  has 
been  set  in  the  position  that  has  to  be  read.  Measure- 
ments are  also  dependent  if  any  essential  step  in  making 
them  is  not  repeated  in  successive  determinations  but 
is  assumed  to  have  its  effect  remain  unchanged  during 
the  series.  Thus,  in  making  independent  measure- 
ments by  the  method  of  coincidences  the  scales  of 
length  were  not  held  in  one  position  while  several 
coincidences  were  found,  but  were  reset  after  each 
determination. 

It  is  customary  to  use  the  term  independent  only 
for  measurements  that  are  at  the  same  time  neither 
dependent  nor  conditioned. 

How  can  density  be  determined  by  a  direct  measure- 
ment? If  so  determined  would  the  mass,  density,  and 
volume  of  an  object  be  independent,  dependent,  or  con- 
ditioned? 

Harmony  and  Disagreement  of  Repeated  Measure- 
ments.— If  the  same  object  is  measured  several  times 
in  succession  the  measurements  will  in  general  differ 
from  one  another,  but  the  differences  will  tend  to 
be  small  under  two  different  circumstances:  (a)  if 
the  quantity  is  of  a  sharply  defined  character,  and 
(6)  if  the  method  of  measurement  is  coarse  or  crude. 
Thvio,  if  a  length  of  woolen  cloth  were  compared  with 
an  accurate  millimetre  scale  it  would  probably  be 
found  difficult  to  measure  it  the  same  twice  in  succes- 


84  MEASUREMENTS  AND  ERRORS 

sion;  but  if  the  same  length  of  steel  rail  were  clamped 
on  rigid  supports  and  measured  at  a  constant  tempera- 
ture it  might  be  hard  to  obtain  two  measurements  that 
would  differ.  The  length  of  one  object  would  be  a 
poorly  defined  quantity;  that  of  the  other,  a  sharply 
defined  quantity.  Although  sharpness  of  definition 
of  the  quantity  to  be  investigated  means  that  repeated 
measurements  will  tend  to  harmonize  closely,  it  should 
be  carefully  noted  that  accuracy  of  the  methods  or 
means  of  measurement  has  just  the  opposite  result; 
it  is  the  rough  methods  of  measurement  that  make 
repeated  determinations  identical  with  one  another, 
and  the  refined  methods  that  show  discrepancies.  Two 
similar  1-lb.  weights  may  appear  to  have  precisely 
the  same  mass  on  a  rough  balance  and  yet  differ  when 
weighed  on  a  more  carefully  constructed  one.  If  the 
heavier  weight  should  then  be  filed  down  just  enough 
to  make  it  equal  to  the  lighter  one  a  test  with  a  delicate 
chemical  balance  might  show  not  only  that  the  two 
were  still  unequal  but  even  that  one  of  them  alone 
would  not  weigh  the  same  amount  twice  in  succession. 

The  general  statement  can  be  made,  then,  that  an 
accurately  defined  quantity  or  a  coarse  method  of 
measurement  will  result  in  a  series  of  determinations 
being  harmonious  or  identical,  while  a  poorly  defined 
quantity  or  an  accurate  method  of  measurement  will 
cause  the  results  to  disagree  or  diverge. 

It  is  a  general  truth  that,  no  matter  how  sharply 
defined  a  quantity  may  be,  the  use  of  the  most  precise 
methods  will  result  in  successive  equally  careful 
measurements  of  it  differing  perceptibly  from  one 
another,  although  the  differences  may  be  very  small. 


MEASUREMENTS  AND  ERRORS  85 

Thus,  the  most  accurate  possible  determinations  of  the 
length  of  a  national  prototype  metre  would  undoubtedly 
differ  by  several  tenths  of  a  micron. 

It  necessarily  follows  that  the  true  magnitude  of  a 
measured  quantity  is  always  unknown;  and  the 
various  approximations  to  it  need  to  be  summarized, 
for  actual  use,  by  their  average  or  by  some  other 
representative  value. 

Errors  of  Measurement. — The  error  of  a  measure- 
ment is  the  amount  by  which  it  differs  from  the  true 
value  of  the  quantity  which  is  measured.  If  the  true 
value  is  always  unknown  the  error  must  likewise  be 
unknown.  Such  errors,  however,  can  be  discussed 
theoretically,  and  in  this  way  much  can  be  learned 
about  the  best  manner  of  dealing  with  them. 

Classification  of  Errors. — Errors  are  classified  as 
constant  and  accidental,  and  what  are  known  as  mis- 
takes really  belong  in  a  separate  class  by  themselves. 
Constant  errors  affect  all  the  measurements  of  a  series 
in  the  same  manner  or  in  the  same  direction.  Accidental 
errors  make  one  measurement  a  trifle  too  large  and 
another  too  small,  but  do  not  tend  to  bias  the  average 
result.  Mistakes  are  occasional  errors  that  are  due 
to  a  lack  of  mental  alertness  on  the  part  of  the  observer. 

ERRORS 

CONSTANT : 

THEORETICAL:     usually  calculable,  as  the  faulty  length  of 

a  linear  scale,  due  to  its  expansion  from  heat. 
INSTRUMENTAL:     due  to  faulty  graduation  or  adjustment 

of  an  instrument. 
PERSONAL:     some  persons  have   a   constant  tendency   to 

estimate  the  instant  of  an  occurrence  a  little  too  early; 

others  a  little  too  late. 


86  MEASUREMENTS  AND  ERRORS 

ACCIDENTAL: 

INSTRUMENTAL:  due  to  varying  external  influences,  "play" 
of  moving  parts,  inconstant  sensitiveness,  etc. 

PHYSIOLOGICAL:  the  senses  of  sight,  touch,  etc.,  have  a 
limit  of  sensitiveness,  and  this  limit  is  not  always  the 
same. 

PSYCHOLOGICAL:  very  likely  the  deductions  about  the 
outside  world  that  result  from  the  effect  of  sense-impres- 
sions on  the  mind  do  not  correspond  to  the  latter  so 
closely  as  to  be  absolutely  free  from  irregular  variations 
when  dealing  with  minute  quantities. 
MISTAKES  : 

MANIPULATIVE:    doing  the  wrong  thing. 

OBSERVATIONAL:    observing  the  wrong  thing. 

NUMERICAL:  recording  the  wrong  numbers;  it  is  especially 
important  to  guard  against  focussing  the  attention  so 
closely  on  the  minute  part  of  a  measurement  (e.  g.,  the 
estimation  of  tenths  of  the  smallest  scale-division)  that 
a  mistake  is  mads  in  recording  the  figures  that  express 
the  larger  part  of  it. 

Accidental  and  Constant  Errors. — The  difference  be- 
tween constant  errors  and  accidental  errors  can  be 
easily  understood  from  the  analogy  of  shots  fired  at 
a  target.  In  the  diagram  the  average  position  or 
centre  of  clustering  has  a  con- 

^^       .  •  •      stant  tendency  downward  and 

^\    ..**•*          "  to  the  ri§ht  of  the  bull's-eye, 
*S5**  •  while  the  individual  shots  have 

*  •          *  • 

•  *t         •  accidental      tendencies      which 

•*  carry  them  to  one  side  of  this 

average    position    as   much    as 

to  the  other.  Any  single  shot  can  be  considered  to 
have  a  total  error  which  is  the  vector  sum  or  geomet- 
rical combination  of  its  own  accidental  error  plus  the 
constant  error  of  the  whole  group.  Physical  measure- 


MEASUREMENTS  AND  ERRORS  87 

ments  are  target  shots  in  which  the  centre  of  clustering 
can  be  found  but  in  which  the  position  of  the  bull's- 
eye  is  unknown.  Constant  errors  can  be  avoided  or 
reduced  to  a  minimum  by  the  help  of  theoretical  knowl- 
edge (effect  of  gravity  in  deflecting  the  shots  down- 
ward), and  by  changing  observers,  methods,  and 
conditions  (repeating  the  shots  when  the  wind  blows 
to  the  left  instead  of  to  the  right),  and  above  all  by 
judgment,  experience,  care,  and  alert  hunting  for  all 
possible  sources  of  error.  Accidental  errors  are  much 
more  easily  investigated,  the  chief  problems  from  the 
standpoint  of  physical  science  being  the  determination 
of  the  centre  of  the  cluster  and  the  measurement  of  the 
degree  of  scattering  which  takes  place  around  it. 

State  what  sources  of  error  were  present  when  you 
performed  each  of  the  experiments  mentioned  in  the 
following  list.  Mark  them  in  such  a  way  as  to  show 
which  ones  gave  rise  to  constant  errors,  and  which 
ones  caused  accidental  errors.  Were  there  any  mis- 
takes? 

The  measurement  of  your  span. 
The  measurement  of  an  irregular  area,  first  method. 
The  measurement  of  the  density  of  an  irregular  solid. 
The  measurement  of  the  sine  of  an  angle  that  has  a  given 
tangent. 

The  experimental  determination  of  TT. 
The  calculation  of  e~x<i  for  different  values  of  x. 
The  use  of  the  formula  1/(1  +  6)  =  1  -  5. 
The  "black-thread"  determination. 

Errors  and  Variations. — The  theory  of  accidental 
errors  runs  closely  parallel  with  the  theory  of  variation 
in  natural  objects,  so  that  a  statistical  investigation 


88  MEASUREMENTS  AND  ERRORS 

of  the  properties  of  several  objects  of  the  same  kind, 
or  variates  as  they  are  technically  called,  furnishes  a 
good  illustration  of  many  of  the  facts  which  could 
otherwise  be  obtained  only  by  the  more  tedious  study 
of  the  variations  in  repeated  measurements  of  the 
same  object. 

Measurement  of  Variates. — Examine  the  vernier  cal- 

iper,  and  if  it  has  an  extra  scale  that  reads  backwards, 

or  one  scale  for  internal  measurements  and  another 

for  external  ones  decide  which  scale 

>  reads  the  internal  distance  between 

cm         frequency  . 

the  jaws  01  the  caliper,  and  which 


1.56 

1.58 
1.59 
1.60 
1.61 
1.62 
1.63 
1.64 
1.65 
1.66 
1.67 
1.68 


point    marks    its  zero    when    the 
jaws    are    closed.     See    that  you 
/  understand   the  vernier  and   have 

//  no  trouble  in  reading  it  at  any  set- 

/////  /////  /         ting. 

Measure  the  length  of  100  seeds 
or  other  variates  of  the  same  kind. 
Make  a  few  preliminary  measure- 
ments to  find  their  general  range 
and  then  make  a  table  like  the 
above,  recording  each  different  length  and  the  number 
of  times  that  it  occurs.  If  an  extreme  value  is  found 
later  it  may  be  noted  anywhere  at  the  beginning  or 
end  of  the  table,  as  shown  here. 

Plot  a  graphic  diagram  in  which  the  measurements 
in  centimetres  are  laid  off  along  the  x-axis  and  the 
frequency  of  each  length  is  represented  by  the  height 
of  a  corresponding  ordinate.  Connect  the  points  of  the 
diagram  by  a  broken  line  and  notice  that  the  curve 
is  highest  in  the  middle  and  slopes  downward,  more 
or  less  uniformly,  toward  the  ends. 


STATISTICAL  METHODS  89 


XI.   STATISTICAL  METHODS 

Frequency  Distributions. — A  tabulation  of  a  set  of 
measurements  that  shows  how  many  times  each  ob- 
served value  .occurs  is  said  to  give  the  "  frequency 
distribution"  of  the  measurements,  and  a  graphic 
diagram  such  as  was  drawn  in  the  preceding  exercise 
is  known  as  a  frequency  polygon.  A  slightly  different 
method  of  drawing  essentially  the  same  diagram  is  by 
using  a  length  on  the  x-axis,  instead  of  a  point,  to 
represent  what  is  called  the  " class-interval"  of  the 


_n 


n  n 

1-56    l$8 


156  Ij&o  1.S4  1.68  156    1JI     IW    U2     164    t£6     168 

frequency  distribution,  or  least  difference  between 
successive  values  of  the  measurement.  Rectangles  are 
then  constructed  on  the  short  base  lines  so  that  their 
height  represents  the  frequency  of  the  corresponding 
measurement.  This  type  of  figure  is  called  a  "  histo- 
gram," and  will  be  seen  to  be  practically  the  same  as 
the  frequency  polygon. 

Class  Interval. — If  the  class  interval  is  made  very 
sm*Jl  and  the  measurements  are  very  numerous,  both 
forms  of  diagram  can  be  considered  as  losing  their 
abrupt  changes  until  they  merge  into  two  identical 


90  STATISTICAL  METHODS 

curved  lines.  If  the  class  interval  is  small,  however, 
without  a  sufficiently  great  number  of  measurements 
the  frequency  of  successive  values  will  tend  to  give  a 
series  of  ordinates  like  0,  1,  0,  0,  1,  0,  1,  0,  1,  1,  1,  0, 
1,  0,  0,  1,  0,  0,  0,  0,  1,  0,  and  the  frequency  diagram  will 
no  longer  have  its  characteristic  " mound-like"  shape 
with  the  ordinates  high  in  the  middle  of  the  diagram 
and  dwindling  toward  each  end. 

If  your  frequency  polygon  tends  to  the  nondescript 
type  seen  when  the  class  interval  is  too  small  the 
measurements  of  the  hundred  variates  should  be  re- 
grouped. Make  a  new  table  in  which  the  values  from 
1.55  to  1.65  are  all  called  1.6,  those  from  1.65  to  1.75 
are  considered  as  1.7,  etc.  If  several  measurements 
have  been  recorded  as  1.65  put  about  half  of  them  in 
the  class  1.6  and  the  other  half  in  the  class  1.7.  From 
this  table  plot  both  the  frequency-polygon  and  the 
histogram  that  illustrate  it  graphically. 

Types  of  Frequency  Distribution. — Frequency  distri- 
butions are  classified,  according  to  the  general  shape 
of  the  graphic  diagram,  as  (a)  symmetrical,  (6)  moder- 
erately  asymmetrical,  (c)  very  asymmetrical  or  J- 
shaped,  (d)  bilocular  or  U-shaped,  (e)  rectangular. 

The  symmetrical  type  (a)  is  seen  in  physical  meas- 
urements: the  average  z-value  is  the  most  frequent; 
measurements  that  are  a  given  amount  above  or  below 
the  average  are  less  frequent  than  it  but  of  equal 
frequency  with  each  other;  and  extremely  diverg- 
ent values  do  not  occur.  The  asymmetrical  type  (6) 
is  similar  except  that  the  most  frequent  value  does 
not  lie  in  the  middle  of  the  distribution  and  on  one 
side  of  it  the  frequency  falls  away  more  rapidly  than 


STATISTICAL  METHODS 


91 


on  the  other.  Moderate  asymmetry  is  common  in  all 
statistical  data.  An  example  of  the  J-shaped  type  (c) 
may  be  seen  in  the  distribution  of  wealth  among  any 
population;  the  frequency  of  individuals  with  little 
wealth  is  very  great,  and  with  increasing  wealth  the 
number  of  cases  falls  off  until  it  reaches  a  vanishing 
point.  The  curious  U-shaped  type  (d)  is  seen  where 


Q 


there  are  tendencies  toward  both  extremes,  or  the  centre 
is  a  position  of  unstable  equilibrium;  and  the  rectangu- 
lar type  (e)  occurs  in  purely  mathematical  cases,  such 
as  the  actual  error  of  a  tabulated  logarithm,  which 
is  never  more  than  ±0.5  of  the  unit  in  the  last  decimal 
place  and  has  all  intermediate  values  with  equal 
frequency. 


92  STATISTICAL  METHODS 

The  Probability  Curve.—  The  measurement  of  100 
seeds  will  probably  show  a  moderate  degree  of  asym- 
metry, since  such  objects  do  not  grow  beyond  a  definite 
size  but  do  fall  short  of  it  in  many  cases.  Physical 
measurements,  however,  tend  to  be  above  the  average 
just  as  often  as  they  are  below  it,  and  so  give  the 
symmetrical  form  of  frequency  distribution.  As  the 
measurements  are  made  more  and  more  numerous  the 
frequency  polygon  approaches  more  and  more  closely 
to  the  form  of  the  so-called  probability  curve,  y  =  e~*2, 
which  was  calculated  in  the  lesson  on  logarithms  and 
plotted  in  the  lesson  on  graphic  representation.  In 
some  cases  it  will  appear  drawn  out  relatively  flat 
and  in  others  will  be  very  high  and  narrow,  but  the 
curve  is  the  same  in  all  cases,  except  that  the  scales 
of  ^-values  and  ^/-values  are  condensed  or  spread  out 
to  different  degrees. 

Carry  out  the  binomial  expansion  that  is  given 
below  at  least  as  far  as  n  =  15,  adding  each  two  terms 
in  order  to  obtain  the  term  below  and  between  them. 
Lay  off  a  series  of  equal  intervals  on  the  #-axis  and 
erect  a  series  of  ordinates  proportional  to  the  succes- 
sive terms  of  the  last  polynomial,  in  order.  A  smooth 
curve  through  the  tops  of  the  ordinates  will  give  a 
very  good  approximation  to  the  probability  curve. 


(l  +  l)1  =  1  +  1 

(1  +  I)2  =  1  +  2  +  1 

(1  +  I)3  =  1+3  +  3+1 

(1  +  I)4  =     1+4  +  6  +  4  +  1 

(l  +  l)5  =1  +  5  +  10  +  10  +  5  +  1 

(1  +  Dn  = 


STATISTICAL  METHODS  93 

Representative  Magnitudes. — For  most  scientific 
work,  the  statement  of  a  whole  frequency  distribution 
or  of  each  one  of  a  long  series  of  measurements  would 
be  a  cumbrous  process  and  reading  such  statements 
would  be  a  tedious  task.  Accordingly  it  is  usual  to 
summarize  such  a  set  of  values  by  stating  some  rep- 
resentative value,  such  as  the  average.  In  special 
cases  such  representative  magnitudes  as  the  geomet- 
rical mean,  -\X(aia2.  •  •  -dn),  or  the  harmonic  mean, 
n/(l/ai  +  l/«2  +  ....+  I/On),  or  the  quadratic  mean, 
V[(«i2  +  tf'22  +  •  •  •  +  an2)/n],  have  been  employed,  but 
the  most  usual  one  is  the  arithmetical  mean  or  average, 
(01  +  a2  +  ...  -\-an)/n.  The  median,  which  is  an  if 
ai,  a 2  .  . .  «2?i— i  are  arranged  in  order  of  size,  is  frequently 
useful  as  a  representative  value,  as  is  also  the  mode 
or  modal  value,  which  is  simply  the  value  that  occurs 
with  the  greatest  frequency. 

The  Average. — The  average  is  obtained  by  adding 
together  a  set  of  values  and  dividing  the  sum  by  the 
number  of  values.  Thus  the  average  of  the  five  values 
3,  3,  4,  5,  10,  is  one-fifth  of  their  sum,  or  5.  The 
average  of  the  measurements  given  in  the  table  at  the 
end  of  the  preceding  lesson  is  (2  X  1.68  +  6  X  1.67  + 
3  X  1.66  .  .  .)/(2  +  6  +  3  .  .  .),  or  8686/53  or  1.639. 

The  Median. — The  median  is  obtained  by  choosing 
such  a  value  that  half  of  the  other  values  exceed  it 
and  half  are  below  it.  If  the  numbers  are  arranged 
in  numerical  order  in  a  column  the  number  that  is 
h^lf  way  down  the  column  is  the  median.  Otherwise 
it  may  be  found  by  crossing  off  the  largest  number  and 
the  smallest,  and  repeating  the  process  until  only  one 
is  left.  If  two  numbers  are  left,  as  will  be  the  case  if 


94 


STATISTICAL  METHODS 


there  are  an  even  number  of  measurements,  the  number 
half-way  between  them  can  be  taken  as  the  median, 
but  in  physical  or  statistical  data  it  usually  happens 
that  the  two  remaining  numbers  are  the  same. 

Find  the  median  of  3,  3,  4,  5,  10.  What  is  the 
median  of  7.4,  6.8,  7.3,  7.3,  7.2,  7.1,  7.2?  Ans:  7.2. 
Find  the  median  of  the  measurements  that  were 
averaged  in  the  preceding  paragraph. 

The  Mode. — The  mode  is  the  most  frequent  value. 
Thus  in  the  set  of  values  3,  3,  4,  5,  10  the  mode  is  the 
number  that  occurs  twice,  namely  3.  In  the  illustration 
of  the  measurement  of  variates  the  mode  is  1.64,  the 
length  that  was  found  most  often. 

Choice  of  Means.— If  a  frequency  distribution  is  of 
the  symmetrical  type  the  average,  mode,  and  median 
will  all  be  the  same. 

Where  there  is  a  moderate  degree  of  asymmetry  the 
median  will  come  nearer  to  the  mode  than  the  aver- 
age does,  as  is  shown  in  the  following  diagram.  The 


Mo  Me  Av 


mode  is  easily  seen  to  be  the  most  probable  value.  If 
a  seed  is  taken  at  random  from  the  set  whose  measure- 
ments are  given  above  it  is  more  likely  to  measure 
1.64  cm.  than  any  other  amount.  The  mode  has  one 
decided  disadvantage,  however,  in  that  it  cannot 


STATISTICAL  METHODS  95 

always  be  chosen  from  a  given  frequency  distribution. 
For  example  the  mode  cannot  be  obtained  from  a 
series  like  3,  4,  4,  5,  6,  6,  7,  8,  except  by  assuming 
that  the  values  follow  the  probability  law,  and  fitting 
a  probability  curve  to  them  as  accurately  as  possible 
by  a  " black  thread"  method;  the  position  of  the 
top  of  this  theoretical  curve  can  then  be  determined. 
Where  there  is  a  long  series  of  measurements  the  mode 
is  sometimes  calculated  from  the  formula,  indicated 
on  the  above  diagram,  that  me  —  mo  =  2  (av  —  me), 
or  that  the  median  lies  J  of  the  way  from  the  average 
to  the  mode. 

The  median,  like  the  mode,  is  easy  to  determine. 
It  can  be  used  in  two  classes  of  cases  where  the  average 
cannot  be  determined.  One  is  in  case  measurements 
have  been  tabulated  with  indefinite  terminal  classes, 
e.  g.,  "less  than  10  mm.,  7  cases;  10  to  12  mm.,  4  cases; 

12  to  14,  5;  14  to  16,  3;  above  16,  2  cases."     Here  the 
median  is  the  class  "between  10  and  12,"  say  11  mm., 
and  the  mode  is  probably  the  class  "12  to  14,"  say 

13  mm.,  but  the  average  cannot  be  found  on  account 
of  nine  of  the  numerical  values  not  being  stated;    the 
best  that  could  be  done  would  be  to  guess  that  the 
average  was  a  little  less  than  the  median.    The  other 
case  is  where  quantities  can  be  arranged  in  numerical 
order  but  are  difficult  to  measure  individually.     Thus 
it  may  be  difficult  or  impossible  to  gauge  the  scholar- 
ship of  a  student  in  accurate  numerical  terms,  but  if  a 
gro^p  of  students  can  be  arranged  in  order  of  scholar- 
ship the  median  can  be  determined  without  difficulty. 
Furthermore  the  median  has  an  advantage  over  most 
other  representative  magnitudes  in  that  it  is  not  affected 


96  STATISTICAL  METHODS 

by  inverting  the  unit  of  measurement.  The  median  of 
a  number  of  prices  will  be  the  same  whether  they  are 
given  as  cents  per  dozen  or  as  dozens  per  dollar.  Of 
a  group  of  different  velocities  the  same  one  will  be 
picked  out  by  choosing  the  median  whether  the  numer- 
ical values  are  expressed  in  miles  per  hour  or  in  minutes 
per  mile.  It  is  easy  to  see  that  this  will  not  be  the 
case  if  the  average  is  used.  The  median,  however,  is 
not  quite  as  good  a  representative  value  as  the  average, 
in  case  they  are  different,  for  a  series  of  measurements 
that  have  all  been  made  with  equal  care.  Curiously 
enough,  however,  the  more  extensive  a  series  of  meas- 
urements the  more  likely  it  is  to  show  that  the  individ- 
ual measurements  are  not  equally  trustworthy  but  may 
be  grouped  in  different  classes  according  to  their  rela- 
tive scattering.  It  is  in  such  cases  that  the  median  is 
a  much  better  representative  figure  than  the  average, 
for  the  average  is  influenced  by  an  unduly  large  or  small 
measurement  just  in  proportion  to  the  aberration  of 
the  latter,  while  as  long  as  a  measurement  is  above  the 
median  it  makes  no  difference  how  far  above  it  may 
be,  its  effect  is  no  greater  than  that  of  any  other  single 
value  (Compare  the  average  and  the  median  of  3,  3, 
4,  5,  10,  with  those  of  3,  3,  4,  5,  30.) 

Find  the  average,  the  mode,  and  the  median  of  your 
measurements  of  100  variates.  What  is  the  ratio  of 
mode  minus  median  to  median  minus  average? 

Deviations. — Of  almost  as  much  importance  as 
finding  the  best  representative  value  for  a  series  of 
measurements  is  the  determination  of  how  closely 
they  cluster  around  it  or  how  widely  they  scatter  from 
it.  This  will  help  to  furnish  information  in  regard  to 


STATISTICAL  METHODS  97 

the  accuracy  of  the  measurements,  and  hence  also  in 
regard  to  the  accuracy  of  the  instruments  and  methods 
employed  in  making  them;  it  will  also  be  useful  in 
comparing  and  combining  determinations  made  at 
different  times  or  by  different  observers.  The  arith- 
metical difference  between  any  single  measurement 
and  the  average,  or  other  representative  magnitude,  is 
known  as  the  "deviation"  of  that  measurement.  It 
is  not  the  same  as  the  error  of  the  measurement,  for 
the  error  may  have  a  constant  component  that  affects 
all  of  the  measurements  equally;  but  it  may  be  con- 
sidered as  the  accidental  error  or  accidental  component 
of  the  total  error.  Deviations  give  no  positive  indi- 
cation of  any  constant  errors  that  may  be  present. 

If  each  one  of  a  series  of  independent  measurements 
is  as  trustworthy  as  any  other  it  can  be  shown  mathe- 
matically that  their  best  representative  value  is  their 
arithmetical  mean,  or  average;  and  accordingly  the 
average  is  the  figure  that  is  almost  invariably  used. 

Copy  the  two  following  columns  of  figures,  find  the 
average  of  each  set,  and  write  after  each  measure- 
ment its  deviation  from  the  average  27  35  27  36 
of  the  figures  in  the  same  column,  27.34  27^38 
marking  it  with  a  minus  sign  if  the  27  33  27  37 
value  is  less  than  the  average,  and  27.34  27.32 
with  a  plus  sign  if  in  excess  of  the  27  33  27  31 
average.  27 .34  27 . 30 

Although  both  averages  are  the  same     ^7  '34       97 '  Q« 
it  is    evident    that    the    first    set  of 
measurements  must  have  been  made  by  a  more  trust- 
worthy   instrument,    observer,    or   method   than    the 
second.     If  the  averages  had  not  been  of  the  same 


98  STATISTICAL  METHODS 

value  the  first  one  would  undoubtedly  have  been 
entitled  to  more  confidence,  other  things  being  equal, 
than  the  second. 

Add  each  column  of  deviations  and  verify  the  fact 
that  the  algebraical  sum  of  the  deviations  from  the  aver- 
age is  always  zero.  If  their  sum  is  zero  their  average 
will  also  be  zero,  so  that  when  an  " average  deviation" 
is  spoken  of  the  term  means  not  the  average  of  the 
algebraical  deviations  but  the  average  of  the  positive 
arithmetical  values  of  all  of  them. 

Average  by  Symmetry. — The  foregoing  property  sug- 
gests an  easy  method  of  finding  the  average  in  simple 
cases :  If  a  number  can  be  so  chosen  that  the  individual 
measurements  are  symmetrically  grouped  around  it 
the  sum  of  the  positive  deviations  will  equal  the  sum 
of  the  negative  deviations  and  the  number  will  be  the 
required  average. 

What  is  the  average  of  115  and  119?  Am:  117, 
because  it  makes  the  sum  of  the  deviations  (+2  and 
—  2)  equal  to  zero. 

Find  the  average  of  3,  6,  9,  12,  15  by  the  method  of 
symmetry. 

What  is  the  average  of  16,  18,  20,  22?  Of  14  and  17? 
Of  12,  14,  17,  19?  Of  126.8  and  127.4?  Of  121  and 
141?  Of  161  and  191?  Of  198,  199,  203?  Of  8,  10£, 
11J?  What  is  the  value,  to  the  nearest  whole  number, 
of  the  average  of  117,  116,  117?  (Suggestion:  Is  the 
average  above  or  below  116.5?)  What  is  the  exact 
average  of  17,  18,  18,  19,  19,  20? 

Average  by  Partition. — Before  leaving  the  subject  of 
the  average  it  should  be  noted  that  there  is  no  need  of 
adding  the  entire  numerical  values  if  they  contain  a 


STATISTICAL  METHODS  99 

part  in  common.  Thus  in  either  of  the  columns  of 
figures  on  page  97  the  average  is  obtained  by  writing 
the  first  three  figures,  which  are  common  to  all  values, 
and  the  average  of  the  last  figure. 

Quartiles. — Just  as  the  middle  number  of  a  series 
arranged  in  ascending  or  descending  order  of  magnitude 
is  called  the  median,  or  half-way  value,  so  the  middle 
one  of  the  numbers  that  lie  below  the  median  is  called 
the  lower  quartile,  or  quarter- way  value;  and  the 
median  of  the  numbers  that  are  greater  than  the  med- 
ian is  known  as  the  upper  quartile,  or  three-quarter- 
way  value.  The  quartile  abscissae  are  laid  off  on  the 
base  line  of  the  diagram,  on  page  94,  at  qi  and  g3.  It 
should  be  carefully  noticed  that  their  ordinates,  to- 
gether with  the  median  ordinate,  divide  the  whole 
area  of  the  frequency  diagram  into  four  equal  parts. 
If  there  is  any  difficulty  in  understanding  this  turn 
to  the  histogram  on  page  89,  and  notice  that  the  area 
indicates  the  total  number  of  measurements.  If  half 
of  them  lie  below  the  median,  by  definition,  and  half 
above  there  can  be  no  trouble  in  realizing  that  the 
median  is  the  abscissa  whose  ordinate  bisects  the  area 
of  the  figure.  Of  course  the  same  thing  is  not  true  of 
the  average,  except  when  average  and  median  happen 
to  have  the  same  value. 

What  are  the  quartiles  of  the  series  3,  4,  4,  5,  6,  8,  11? 

Find  the  median  and  quartile  values  of  the  set  of 
numbers  28,  29,  31,  36,  31,  30,  35,  33,  32,  36,  29,  28,  32. 

Semi-Interquartile  Range. — The  numerical  distance 
between  the  lower  quartile  and  the  upper  quartile  is 
called  the  interquartile  range.  In  the  series  7,  8,  10, 
13,  19,  22,  27,  the  quartiles  are  8,  and  22,  and  their 


100  DEVIATION  AND  DISPERSION 

difference,  14,  is  the  range  between  quartiles,  or 
interquartile  range.  Half  of  this,  or  7,  is  called  the 
semi-interquartile  range;  notice  that  it  is  the  average 
of  the  two  distances  from  median  to  quartile.  It  is 
sometimes  used  as  a  measure  of  the  scattering  or 
clustering  of  a  set  of  observations,  and  its  theoretical 
importance  will  be  seen  a  little  later. 

Find  the  semi-interquartile  range .  of  each  of  the 
columns  of  figures  on  page  97.  For  the  first  column: 
if  the  five  smaller  values  are  considered  as  lying  below 
the  median,  that  is,  if  the  median  is  considered  to 
occupy  no  numbers  in  the  middle  of  the  column  the 
lower  quartile  will  be  27.34;  if  only  four  values  are 
considered  as  lying  below  the  median,  that  is,  if  the 
median  is  considered  to  include  two  numbers  in  the 
middle  of  the  column  the  lower  quartile  will  be  27.335; 
as  the  median  is  actually  to  be  considered  as  one  single 
number  it  will  be  satisfactory  to  say  that  the  lower 
quartile  is  half-way  between  these  two  values,  say  27.338. 
Similarly,  the  upper  quartile  may  be  taken  as  27.342. 
The  second  column  should  be  treated  in  thesame  manner. 


XII.   DEVIATION   AND   DISPERSION 

Apparatus. — Vernier  caliper;  variates;  slide  rule. 

Characteristic  Deviations. — Just  as  the  use  of  an 
average  or  other  representative  magnitude  makes  it 
unnecessary  to  state  the  separate  measurements  from 
which  it  is  derived,  so  a  statement  of  all  the  deviations 
from  the  average  can  be  replaced  by  a  single  char- 
acteristic deviation.  A  set  of  measurements  can  then 
be  summarized  by  two  numbers,  and  it  is  customary 


DEVIATION  AND  DISPERSION  101 

to  write'  such  a  result  in  the  form  a  =*=  d,  where  the 
first  number  gives  the  average  value  of  the  quantity 
measured  and  the  second  expresses  the  limiting  dis- 
tances above  and  below  the  avera£<^  winch  include 
half  of  the  measurements  or  which  mark  ofTin  some 
other  way  an  amount  of  deviation  4£hich;  -Jg  ^ftafjsctfK 
istic  for  the  set  of  measurements. 

Total  Range. — The  simplest  way  of  summarizing  the 
deviations  of  a  series  of  measurements  is  by  stating 
their  total  range,  or  the  algebraical  difference  between 
the  smallest  and  largest.  Obviously  the  extreme  range 
of  the  measurements  themselves  will  also  give  the  same 
result.  Thus,  in  the  two  columns  of  measurements 
that  have  been  considered  above  the  total  range  is 
.02  for  the  first  and  .08  for  the  second.  Unfortunately, 
this  is  also  the  worst  method  of  obtaining  a  character- 
istic deviation,  for  the  total  range  depends  upon  only 
two  of  the  measurements  of  the  series;  and  those  two 
are  the  least  satisfactory  ones,  because  repetitions  of 
the  series  of  measurements  would  undoubtedly  show 
a  considerable  fluctuation  in  the  largest  and  smallest 
values,  while  the  most  closely  clustered  values  would 
hardly  be  changed  at  all. 

Average  Deviation. — A  better  index  of  the  amount  of 
scattering  is  given  by  the  average  deviation.  This  is  the 
average  of  the  positive  arithmetical  values  of  the  devia- 
tions. The  true  algebraical  average  of  the  deviations 
cannot  be  used  if  they  have  been  calculated  from  the 
average  measurement  because,  as  has  been  shown,  its 
value  is  always  zero. 

The  average  of  the  positive  values  of  the  deviations 
is  always  smallest  when  the  deviations  have  been  cal- 


102  DEVIATION  AND  DISPERSION 

ciliated  from  the  median  measurement,  and  it  is  in  con- 
nection with  the  median  that  it  is  generally  used. 

Find  the  average  and  the  median  of  the  numbers  3, 
3,  4;  5,  10  "What  is  the  average  deviation  from  the 
median?  Wna^  *s  ^  from  the  average? 

Starfdajd'Deyiatioii.—  The  typical  deviation  that  is 
most  frequently  used  for  statistical  purposes  is  the 
standard  deviation.  This  is  the  square  root  of  the  quo- 
tient of  the  sum  of  the  squares  of  the  deviations  from 
the  average  divided  by  one  less  than  the  number  of 
statistical  values.*-  If  there  are  n  quantities  whose 
deviations  from  their  average  are  respectively  di,  d^ 
dx,  ...  dn,  the  standard  deviation  of  those  quantities 
is  the  value  of 


n-l 

The  standard  deviation  is  a  special  case  of  the 
mean-square  deviation,  being  in  fact  the  mean-square 
deviation  from  the  average,  or  approximately  the 
quadratic  mean  of  the  deviations  from  the  average. 
It  is  also  sometimes  called  the  mean  deviation  and  the 
mean-square  deviation,  but  as  the  term  mean  deviation 
is  also  used  for  what  we  have  denned  as  the  average 
deviation  it  is  much  better  to  avoid  the  use  of  the  word 
mean  altogether  in  connection  with  a  deviation. 

Dispersion. — The  measure  of  deviation  which  is  used 
for  physical  measurements  in  almost  all  cases  is  that 

*  To  be  strictly  accurate,  the  standard  deviation  of  the  statis- 
tician is  V  2d2  /  V  n.  The  reasons  for  using  n-l  will  be  found  in 
the  mathematical  works;  here  it  need  only  be  noticed  that  the 
denominator  is  diminished  because  the  "numerator  is  smaller 
than  the  sum  of  the  squares  of  the  true  errors  (page  124).  Of 
course,  if  n  is  fairly  large,  the  distinction  is  unnecessary. 


DEVIATION  AND  DISPERSION  103 

particular  form  of  characteristic  deviation  which  is 
called  the  dispersion.  It  is  approximately  two-thirds 
of  the  standard  deviation,  or,  more  exactly 

*  +  d?  +  .    .  +  d,*. 


.6745 


n-l 


Significance  of  the  Dispersion. — An  important  char- 
acteristic of  the  dispersion  is  that  for  a  series  of  meas- 
urements which  is  extensive  enough  to  follow  the  law 
of  the  probability  curve  this  typical  value  can  be 
shown  mathematically  to  express  exactly  the  same 
limits  above  and  below  the  average  as  are  given  with 
respect  to  the  median  by  the  semi-interquartile  range. 
If  measurements  follow  the  law  of  the  probability 
curve  the  median  and  the  average  will  coincide;  and 
the  upper  and  lower  quartiles  include  just  half  of  the 
total  number  of  measurements  within  the  space  between 
them,  as  was  seen  in  connection  with  the  curve  on  page 
67.  In  other  words,  the  right-hand  half  of  the  histo- 
gram will  be  bisected  by  the  ordinate  corresponding  to 
the  positive  value  of  the  dispersion,  and  the  correspond- 
ing abscissa  will  be  the  median  value  of  all  the  positive 
deviations.  Since  the  left-hand  half  of  the  curve  is 
likewise  bisected  by  the  negative  value  of  the  devia- 
tion, and  the  graphic  diagram  is  symmetrical  with 
respect  to  the  ?/-axis,  it  follows  that  the  dispersion  is 
the  median  of  the  absolute  values  of  all  the  deviations, 
or  that  any  single  deviation  is  as  likely  to  be  less  than 
the  dispersion  as  it  is  to  be  greater.  The  dispersion, 
jv  ith  its  plus-and-minus  sign  marks  off  a  small  distance 
around  the  average,  and  it  is  just  as  likely  as  not  that 
any  single  measurement  chosen  at  random  will  be 
within  these  limits. 


104 


DEVIATION  AND  DISPERSION 


Advantage  of  the  Dispersion. — Where  the  number  of 
measurements  in  a  series  is  relatively  small,  say  not 
greater  than  10,  the  dispersion  obtained  by  calculation 
and  the  semi-interquartile  range  obtained  by  picking 
out  the  quartile  values  will  not  usually  be  of  the  same 
value,  on  account  of  the  measurements  not  being 
sufficiently  numerous  to  follow  the  laws  of  probability 
very  closely.  Even  in  this  case,  however,  the  value 
given  by  the  formula  is  preferable  to  half  the  differ- 
ence between  the  quartiles  because  the  former  is  ob- 
tained from  all  the  measurements  of  the  series  and  the 
latter  from  only  two.  Where  there  are  as  many  as  ten 
measurements  of  a  physical  magnitude  it  is  usually 
found  that  the  two  values  will  not  differ  by  more  than 

15  or  20  per  cent.,  and 
for  rough  preliminary 
measurements  the 
serni-i  nterquartile 
range  is  almost  as  satis- 
factory as  the  disper- 
sion and  can  be  ob- 
tained much  more 
readily. 

Calculation  of  the 
Dispersion.  —  Measure 
ten  variates  taken  at 
random  from  the  100 
that  were  measured 
before,  and  tabulate 
them  in  a  column 
headed  v.  Find  the 
average  and  write  the 


V 

d 

d2 

1.82cm 

33 

1089 

1.85 

63 

3969 

1.85 

63 

3969 

1.78 

7 

49 

1.82 

33 

1089 

1.84 

53 

2809 

1.78 

7 

49 

1.62 

167 

27889 

1.81 

23 

529 

1.70 

87 

7569 

17.87  (10 

49010.  .  .Z(d2) 

1.787  .  .av. 

49010  =  4.6903 


log  9 


.  9542 


2)3.7361 

1.8680 

%.6745  =    1.8290 

1  .  6970  =  log  D 
D  =  49.78 


DEVIATION  AND  DISPERSION  105 

deviations  without  decimal  points  in  an  adjacent  column 
d.  In  a  third  column  write  the  respective  values  of 
d2,  using  the  table  of  squares  on  page  141  to  obtain 
them.  Then  find  the  sum  of  the  squares,  divide  it  by 
n  —  1,  9  in  this  case,  and  multiply  the  square  root  of 
the  quotient  by  .6745.  The  product  will  be  the  dis- 
persion. 

If  the  numbers  in  column  d  are  expressed  in  thou- 
sandths of  a  centimetre  the  dispersion  will  also  come 
out  in  thousandths;  thus,  the  value  of  D  shown  in  the 
illustration  means  .04978  cm. 

Rule  for  Accuracy  of  the  Average. — In  determining 
how  many  figures  of  the  average  are  to  be  retained  as 
significant  it  is  best  to  follow  the  rule  that  at  least  half 
of  the  deviations  should  be  greater  than  3.  In  the  illus- 
trative example  it  will  be  noticed  that  keeping  three 
decimal  places  in  the  average  has  made  more  than  half 
of  the  deviations  greater  than  30.  In  such  a  case  the 
average  could  obviously  be  rounded  off  to  two  decimal 
places,  1.79,  and  at  least  half  the  deviations  would 
still  be  greater  than  3.  This  will  be  found  to  simplify 
the  calculation  and  the  result  will  not  be  essentially 
different.  In  the  above  example  the  result  would  have 
come  out  4.982  (hundredths  of  course,  this  time,  not 
thousandths),  or  .04982  cm,  which  is  seen  to  agree  with 
the  previous  result  to  three  significant  figures,  and  two 
significant  figures  are  all  that  is  usually  wanted  for 
the  value  of  a  dispersion. 

Use  of  the  Table  of  Dispersions. — The  root  ex- 
traction and  long  multiplication  and  division  should, 
of  course,  never  be  done  by  the  tedious  arithmetical 
process.  Even  the  logarithmic  process  used  in  the 


106  DEVIATION  AND  DISPERSION 

illustration  can  be  avoided  as  follows:  49010  -r-  9  = 
5446;  the  square  root  of  this  will  be  somewhat  more 
than  70;  f  of  70  is  about  48;  in  the  column  headed 
(n  =±  J)2/-67452  of  the  table  on  page  141  find  two  num- 
bers between  which  5446  lies  and  read  the  correspond- 
ing number  in  column  n.  It  is  immediately  seen  to  be 
50,  which  agrees  with  the  previously  obtained  49.78  as 
far  as  the  two  significant  figures  which  are  all  that  is 
required.  As  any  arrangement  of  significant  figures  has 
two  square  roots  (see  page  60)  a  place  for  5446  would 
also  be  found  opposite  16  in  the  table,  but  the  rough 
preliminary  check-calculation  showed  that  the  answer 
should  be  about  48,  so  there  could  be  no  doubt  that 
the  required  answer  is  50  rather  than  16. 

Dispersions  with  the  Slide  Rule. — In  all  following 
work  in  physics  dispersions  are  to  be  calculated  either 
with  the  table,  as  explained,  or  with  the  slide  rule, 
which  makes  the  process  even  easier.  A  special  line 
is  marked  at  6745  on  the  C  scale  so  that  .6745\/a/6 
can  be  obtained  with  a  single  setting  as  soon  as  the  sum 
of  the  squares  of  the  deviations  is  obtained.  Difficulty 
with  the  double  square  root  is  best  avoided  by  setting 
the  end  of  the  C  scale  to  the  approximate  value  of  the 
radical  as  obtained  by  a  rough  mental  calculation;  a 
very  slight  movement  of  the  slide  is  then  all  that  is 
needed  to  make  an  exact  setting. 

Sigma-notation. — The  capital  letter  sigma  (2)  of  the 
Greek  alphabet  is  often  used  by  mathematicians,  pre- 
fixed to  an  algebraical  term,  to  denote  the  sum  of  all 
such  terms;  thus  the  expression  for  the  average 
(ai  +  a2  +  a3  .  .  .  +  an)/n  is  abbreviated  to  the  equiva- 
lent form  (Sa)/w.  In  the  same  way  the  formula 


DEVIATION  AND  DISPERSION  107 

.. +*•  +...+*.• 


is  easier  to  write  in  the  form 
.6745 


and  the  use  of  2(d2)  will  have  been- noticed  previously 
in  the  example  of  the  calculation  of  a  dispersion. 

Rewrite  the  formulae  for  the  harmonic  mean  and 
the  quadratic  mean  (page  93),  using  the  sigma-nota- 
tion.  Write  the  formula  for  (x\  +  z2)2,  using  the  same 
notation. 

If  five  measurements  of  the  quantity  x  give  3,  3,  4, 
5,  10,  what  is  the  value  of  Zz?  If  the  average  is  5 
what  is  the  value  of  2d?  Of  Z(d2)?  Of  S(z2)? 

Dispersion  of  an  Average. — The  average  length  of  10 
variates  has  already  been  calculated.  If  10  more  were 
measured  their  average  would  be  somewhere  nearly 
the  same  as  the  first  one.  If  a  considerable  number  of 
such  averages  had  been  determined  it  would  be  a 
simple  matter  to  determine  their  dispersion,  and  the 
result  would  naturally  be  a  smaller  number  than  the 
dispersion  of  any  set  of  individual  measurements,  since 
an  average  is  a  more  trustworthy  figure  than  a  single 
determination.  In  fact  it  can  be  shown  mathematic- 
ally that  if  ten  equally  good  measurements  are  averaged 
the  single  measurements  will  show  a  variation  which 
is  greater  than  the  variation  of  such  averages  in  the 
proportion  of  \/ 10  to  1,  and  similarly  that  the  dispersion 
of  averages  will  be  \l\/n  as  great  as  the  dispersion  of 
individual  measurements  if  the  latter  are  averaged  in 
groups  of  n.  This  means  that  it  is  not  necessary  to  cal- 
culate several  averages  in  order  to  find  their  dispersion, 


108  DEVIATION  AND  DISPERSION 

for  it  can  be  determined  from  a  knowledge  of  the 
number  of  measurements  that  go  to  make  up  a  single 
average  and  from  the  dispersion  of  these  individual 
measurements.  Thus  the  dispersion  of  the  average,  as 
it  is  called,  of  nine  measurements  is  at  once  seen  to  be 
one-third  as  large  as  the  dispersion  for  single  measure- 
ments, since  the  square  root  of  nine  is  three. 

The  formula  for  the  dispersion  of  an  average  is  easily 
written,  for  if 


then 

r»          A74x 
Da,  =  .6745 


the  second  formula  being  I/  V  n  as  large  as  the  first. 

The  Statement  of  a  Measurement.  —  Tabulate  the 
first  eleven  of  the  twelve  measurements  of  the  wooden 
block  made  with  the  card-board  model  of  a  vernier 
caliper.  Pick  out  their  semi-interquartile  range,  to 
be  used  as  a  rough  value  of  the  dispersion  of  the  indi- 
vidual measurements,  and  divide  it  by  the  square  root 
of  n.  Calculate  the  dispersion  of  the  average  and 
compare  it  with  the  approximation  just  found,  employ- 
ing the  usual  rule  for  comparison:  "divide  the  differ- 
ence by  the  greater  number."  Write  the  thickness  of 
the  block  in  the  form  av.  =±=  Dav,  the  customary  method 
of  stating  a  measurement  of  any  kind.  Divide  DI  for 
your  measurement  of  10  seeds  by  V  10  in  order  to 
obtain  Day  for  them,  and  state  their  measurement  in 
the  same  form. 

Make  eleven  more  measurements  of  the  wooden 
block,  this  time  with  the  vernier  caliper  that  gives 


THE  WEIGHTING  OF  OBSERVATIONS          109 

tenths  of  a  millimetre,  and  treat  them  in  the  same  way 
as  the  previous  eleven.  What  can  you  learn  from 
the  results  obtained  with  the  two  different  pieces  of 
apparatus? 

Relative  Dispersion. — It  has  already  been  shown  that 
in  order  to  see  how  serious  an  error  really  is  it  should 
be  divided  by  the  true  magnitude  of  the  quantity 
measured.  In  the  same  way  it  is  often  found  that  the 
dispersion  itself  does  not  give  as  much  information  as 
the  proportional  dispersion,  or  relative  dispersion,  or 
fractional  dispersion,  as  it  is  also  called;  the  ratio 
of  dispersion  to  representative  magnitude. 

In  the  two  measurements  just  stated,  the  length  of 
a  seed  and  the  thickness  of  the  wooden  block,  divide 
the  dispersion  of  the  average  by  the  average  itself  in 
order  to  obtain  the  relative  dispersion  of  the  average. 
Express  this  either  as  a  decimal  or  as  a  percentage. 
For  which  set  of  measurements  should  you  expect  it  to 
be  the  smaller?  Why?  Does  the  same  relation  hold 
true  for  your  dispersions  of  the  averages  as  expressed  in 
centimetres?  Explain  why. 

XIII.   THE  WEIGHTING  OF  OBSERVATIONS 

Apparatus. — Platform  balance  and  clamp;  set  of 
weights;  vernier  caliper;  aluminum  block;  over-flow  can 
and  catch-bucket  for  measuring  displaced  water;  string 
and  two  spreading  rods;  fine  silk  thread;  slide  rule. 

Necessity  of  Weights  for  Observations. — A  represen- 
tative value  is  often  wanted  for  measurements  which 
are  not  all  equally  trustworthy.  The  accepted  values 
for  such  constants  as  the  maximum  density  of  water, 


110  THE  WEIGHTING  OF  OBSERVATIONS 

the  mechanical  equivalent  of  heat,  the  length  of  the 
true  ohm  of  mercury,  the  velocity  of  light  in  vacuo, 
have  all  been  derived  from  measurements  by  different 
observers  at  various  times,  and  in  general  by  different 
apparatus  and  methods.    Any  of  these  varying  factors 
will  produce  varying  results,   and  one  determination 
can  be  accepted  with  more  confidence  than  another 
and  so  will  be  entitled  to  greater  " weight"  when  it 
necessary  to  decide  upon  a  representative  value. 

Density  by  Different  Methods. — An  example  of  the 
effect  of  different  methods  on  the  determination  of  a 
physical  magnitude  may  be  given  by  the  measurement 
of  the  density  of  a  metal  block.  If  the  mass  is  known 
this  can  be  accomplished  by  mensuration,  by  measuring 
displacement,  or  by  a  measurement  of  buoyant  force. 
According  to  the  Principle  of  Archimedes  the  apparent 
loss  of  weight  of  a  body  immersed  in  a  fluid  is  the  same 
as  the  weight  of  an  equal  volume  of  the  fluid.  If  the 
volume  of  a  metal  block  is  v,  its  weight  w,  and  its 
apparent  weight  in  water  wf,  the  density  can  be  found 
as  the  ratio  of  the  weight,  w,  to  the  loss  of  weight, 
w  —  wf,  supposing  that  the  density  of  water  is  unity; 
or  it  can  be  determined  as  the  ratio  of  the  weight,  w, 
to  the  volume  of  water  that  is  actually  displaced  on 
immersion,  say  v' ;  or  the  block  can  be  measured  with 
a  caliper  and  the  density  calculated  as  m/v. 

Clamp  the  platform  balance  to  the  cross-bar  above 
the  table  in  such  a  way  that  an  object  can  be  weighed 
by  suspending  it  under  the  bar  with  strings  attached 
to  a  spreading  rod  on  one  scale-pan  of  the  balance. 
Use  the  other  spreading  rod  as  a  counterpoise,  and 
make  a  careful  allowance  for  the  fact  that  they  may 


THE  WEIGHTING  OF  OBSERVATIONS          111 

not  exactly  balance.  Attach  the  aluminum  block  to 
the  string  by  a  fine  thread  long  enough  to  allow  it  to 
hang  within  the  empty  overflow  can,  and  weigh  it  as 
accurately  as  possible.  Fill  the  overflow  can  with 
water,  closing  the  spout  with  the  finger-tip;  place  it  in 
position  where  the  aluminum  block  is  to  hang,  with 
the  catch-bucket  under  the  spout;  allow  the  excess 
of  water  to  run  out  of  the  overflow  can ;  then  weigh  the 
catch-bucket  with  its  contained  water,  and  replace  it 
in  position.  Lower  the  aluminum  block  carefully 
into  the  overflow  can  and  weigh  it  while  submerged; 
then  weigh  the  catch-bucket  again  in  order  to  find  out 
how  much  water  was  displaced. 

Find  the  density  of  the  aluminum  block  (a)  by  com- 
paring its  weight  with  the  weight  of  water  actually 
displaced;  (6)  from  the  two  values  w  and  wr;  (c)  by 
measuring  the  block  with  the  vernier  caliper,  com- 
puting its  volume  as  closely  as  possible,  and  applying 
the  formula  for  density,  d  =  m/v.  Report  your  results 
for  comparison  with  those  of  the  other  members  of  the 
class. 

Weights  for  Repeated  Values. — The  simplest  case  of 
weighting  different  observations  is  when  separate  numer- 
ical values  have  each  been  obtained  a  definite  number  of 
times.  Suppose,  for  example,  that  the  density  of  a 
block  of  aluminum  has  been  determined  both  as  2.6 
and  as  2.7,  the  smaller  value  having  been  found  on 
four  occasions  while  the  larger  value  was  obtained  only 
once  in  the  total  of  five  measurements.  The  best 
representative  figure  from  these  data  certainly  would 
not  be  the  number  2.65,  half  way  between  2.6  and  2.7, 
but  ought  to  be  a  number  situated  four  times  as  far 


112  THE  WEIGHTING  OF  OBSERVATIONS 

from  the  least  frequent  measurement,  2.7,  as  from 
the  most  frequent  one,  2.6;  in  other  words,  it  should 
be  the  number  2.62.  Moreover,  this  is  easily  seen  to 
be  the  same  result  as  would  be  obtained  by  taking 
the  average  of  the  five  individual  measurements.  The 
rule  in  such  a  case  is  obviously  to  give  each  numerical 
value  a  weight  proportional  to  the  number  of  times  of 
its  occurrence. 

Find  the  weighted  average  of  the  values  of  a  meas- 
ured length  if  it  was  found  to  be  2.345  cm  in  each  of 
six  trials,  2.350  cm  in  twelve  trials,  and  2.355  in  nine 
trials.  Suggestion :  calculate  the  value  of  2  X  2.345  + 
3  X  2.355  +  4  X  2.350. 

The  Weighted  Average. — The  weighted  average  is 
found  in  any  case  by  considering  that  certain  values 
have  been  obtained  more  frequently  than  others.  In 
the  case  just  discussed  this  was  a  fact,  in  other  cases 
it  is  only  a  supposition  made  to  fit  the  known  or 
estimated  intrinsic  value  of  the  observations. 

If  a  difficult  measurement  had  been  made  by  an  ex- 
perienced student  and  found  to  be  0.35,  while  the  same 
experiment  gave  the  value  0.41  when  performed  by  a 
beginner,  it  might  be  decided  somewhat  arbitrarily  to 
give  the  first  number  twice  the  weight  of  the  second. 
The  process  of  finding  the  weighted  average,  (2  X 
0.35  +  1  X  0.41)/3,  would  then  be  equivalent  to  sup- 
posing that  the  better  measurement  had  been  obtained 
on  two  occasions  but  the  poorer  one  only  once.  If 
a  measurement  of  some  quantity  had  been  found  to  be" 
1.36  when  made  under  unfavorable  circumstances,  and 
1.41  when  made  under  circumstances  that  were  more 
favorable  to  experimentation  it  might  be  considered 


THE  WEIGHTING  OF  OBSERVATIONS          113 

best  to  assign  the  respective  weights  of  1  and  1.5 
to  the  two  values.  The  weighted  average  would  then 
be  (2  X  1.36  +  3  X  1.41)  -f-  5,  or  1.39,  a  figure  which 
will  be  seen  to  be  nearer  to  the  better  value  than  to 
the  poorer  one  in  exactly  the  ratio  of  1  to  1.5. 

Arbitrarily  Assigned  Weights. — The  objectionable 
feature  of  such  an  arbitrary  assignment  of  weights  is 
very  obvious.  The  relative  weights  depend  too  much 
upon  the  judgment  of  the  individual  computer;  further- 
more, it  is  often  difficult  to  avoid  being  influenced  by 
the  fact  that  certain  determinations  vary  more  or  less 
widely  from  the  expected  value,  instead  of  keeping  one's 
judgment  focussed  on  the  quality  of  the  experimental 
work. 

Which  do  you  consider  the  better  method  of  deter- 
mining density,  by  buoyancy  or  by  displacement? 
Choose  what  you  consider  the  best  ratio  for  their 
relative  accuracies  and  find  the  corresponding  weighted 
average,  but  be  careful  not  to  give  extra  weight  to 
either  measurement  on  account  of  its  coming  close  to 
the  third  determination  made  by  calculating  the  volume 
obtained  by  mensuration. 

Weight  and  Dispersion. — Determinations  of  any  care- 
fully measured  magnitude  are  usually  stated  in  the 
form  of  an  average  and  its  dispersion.  If  the  influence 
of  constant  errors  can  be  neglected  it  can  be  shown 
mathematically  that  the  best  value  for  the  measure- 
ment is  obtained  by  weighting  each  determination  in 
inverse  proportion  to  the  square  of  its  dispersion.  Thus, 
iL  one  determination  has  a  dispersion  of  .0040  and 
another  has  a  dispersion  of  .012  the  former  should 
be  given  nine  times  as  much  weight  as  the  latter. 


114  THE  WEIGHTING  OF  OBSERVATIONS 

This  can  be  expressed  in  a  general  formula  by  saying 
that  the  weighted  average  of 


s 


or 

w.  av.  =  S 

but  it  is  much  better  to  learn  the  principle  involved 
than  to  memorize  the  formula. 

Limitations  of  w  =  k/d2.  —  Attention  should  again  be 
directed  to  the  fact  that  weighting  according  to  disper- 
sions takes  no  account  of  the  fact  that  constant  errors 
may  be  present  in  the  given  data.  The  dispersion 
summarizes  only  the  accidental  errors,  and  if  the 
constant  errors  are  greater  than  these  the  weighted 
average  is  no  better  than  the  simple  arithmetical 
average. 

Tabulate  the  determinations,  made  by  the  various 
members  of  the  class,  of  the  density  of  aluminum  as 
found  by  the  effect  of  buoyancy.  Calculate  the  typical 
value  in  the  form  a  =*=  d. 

Find  in  the  same  way  the  average  and  dispersion  of 
the  density  as  determined  by  displacement. 

Calculate  the  weighted  average  of  these  two  data. 

There  are  several  sources  of  constant  error  in  each 
of  the  two  above  methods  of  determining  density. 
State  at  least  four  that  are  common  to  both  methods, 
and  at  least  one  that  influences  one  form  of  experiment 
but  not  the  other. 

Exception  to  the  Rule.  —  The  method  of  weighting  ob- 
servations in  inverse  proportion  to  their  dispersions  is 


CRITERIA  OF  REJECTION  115 

used  for  separate  and  independent  data  whose  relative 
accuracy  is  assumed  to  be  shown  by  their  dispersions. 
Where  two  or  more  series  of  observations,  however, 
are  known  to  have  been  made  with  equally  reliable 
apparatus,  methods,  and  observers  they  should  be 
weighted  merely  according  to  the  number  of  measure- 
ments which  each  comprises,  notwithstanding  that 
their  dispersions  might  indicate  a  very  different  result. 
To  do  otherwise  would  be  to  repudiate  the  principle 
of  the  average,  which  depends  upon  the  fact  that  all 
observations  are  supposed  to  be  equally  trustworthy. 
On  the  other  hand,  when  different  observations  are 
known  to  be  unequally  trustworthy,  even  if  they 
occur  in  the  same  series,  weight  may  be  given  to  the 
fact  that  some  are  closely  clustered  about  an  apparent 
central  position  while  others  diverge  erratically. 

Which  is  to  be  preferred,  the  average  or  the  median, 
for  a  determination,  like  the  one  just  made,  of  density 
by  buoyancy?  Why?  (Refer  back  to  Lesson  XI.) 

XIV.   CRITERIA   OF   REJECTION 

Apparatus. — Slide  rule. 

Observational  Honesty. — When  successive  redeter- 
minations  of  a  quantity  have  been  made  in  the  course 
of  an  experimental  investigation  it  is  to  be  supposed 
that  they  have  all  been  made  with  an  equal  degree  of 
care. 

It  is  important  to  remember  than  an  observation 
snould  never  be  rejected  simply  because  it  is  not  in 
satisfactory  agreement  with  the  other  determinations 
of  the  series.  If  the  experimenter  realizes  that  one  of 


116  CRITERIA  OF  REJECTION 

his  measurements  was  made  under  some  kind  of  a 
handicap  or  under  such  conditions  that  a  faulty  result 
would  be  likely  it  is  permissible  to  cross  out  the  cor- 
responding value  in  his  notes  and  to  omit  it  in  the 
final  consideration  of  the  data,  but  there  must  be  some 
definite  and  satisfactory  reason  for  discarding  it  other 
than  the  fact  of  its  divergence  from  the  expected  value. 
The  temptation,  often  felt  by  the  beginner,  to  omit  or 
"  re-determine  "  a  discordant  result  may  be  very  percep- 
tible, but  absolute  integrity  of  observation  should  be 
cultivated  to  such  a  point  that  the  experimenter  is 
habitually  able  to  feel  a  certain  disinterestedness  in  the 
outcome  of  a  measurement  after  he  has  first  taken 
pains  to  ensure  its  being  as  trustworthy  as  possible. 

Importance  of  Criteria. — Even  with  all  care  to  make 
successive  measurements  equally  accurate  it  often  hap- 
pens that  one  or  more  of  them  show  unduly  large 
deviations  from  the  average.  In  order  to  prevent  these 
values  from  having  an  abnormal  influence  on  the 
representative  value  certain  rules  have  been  formu- 
lated to  determine  whether  they  shall  be  retained  or 
discarded,  for  if  an  observer  merely  used  his  own  judg- 
ment in  deciding  the  question  the  result  would  depend 
too  much  upon  his  own  individuality  and  tempera- 
ment, and  different  observers  would  obtain  different 
results  from  data  identically  the  same,  just  as  in  the 
case  of  arbitrarily  assigned  weights.  In  fact,  the 
rejection  of  a  measurement  is  nothing  more  nor  less 
than  giving  it  a  weight  equal  to  zero. 

Chauvenet's  Criterion. — One  of  the  easiest  to  under- 
stand of  the  various  devices  for  testing  doubtful  obser- 
vations is  known  as  Chauvenet's  criterion  of  rejection, 


CRITERIA  OF  REJECTION  117 

according   to   which   rejectability  is    deter- 
mined as  a  function  of  deviation,  dispersion, 

and  number  of  measurements.  Q  ^Q  Q 

Draw  a  graphic  diagram  from  the  table.  j  499 

Draw  the  ordinates  x  =  10  and  x  =  50  from  2  49.6 

the  base  line  up  to  the  curve.  The  result  will  4  48'  2 

be  the  right-hand  half  of  a  normal  frequency  5  47  2 

polygon,  the  ^-values  corresponding  to  devi-  6  46  0 

ations  and  the  ^/-values  to  the  frequency  of  7  44.6 

their  occurrence.    Remember  that  the  total  9  41^6 

area  of  such  a  curve  corresponds  to  the  num-  10  39  9 

ber  of  deviations ;  in  the  same  way  the  area  be-  j  2  350 

tween  curve  and  base  line  which  is  bounded  14  32.1 

i  A  oft  c\ 

by  any  two  ordinates  represents  the  number  18  24  o 

of  observations  whose  numerical  deviations  20  20.2 

lie  between  those  two  limits.    As  the  disper-  22  16.6 

sion  is  the  same  as  the  median  deviation  it  is  24  13.8 

26  If)  S 

evident  that  the  ordinate  which  bisects  the  28  8'4 

area  (the  ordinate  x  =  10,  for  the  scales  used  30  6  5 

in  this  diagram)  must  have  its  abscissa  nu-  32  4  8 
merically  equal  to  the  dispersion. 

Suppose  another  ordinate  is  so  drawn  as  38  j  8 

to  include  nine-tenths  of  the  area  between  40  i  2 

the  ?/-axis   and  itself,  and  leave  only  one-  42  0  9 

tenth  of  the  area  beyond  it  to  the   right,  44  0.7 

then  the  corresponding  abscissa  would  sim-  48  0'3 

ilarly  have  a  value  that  would  be  exceeded  50  0  2 

by  only  one-tenth  of  the  total  number  of  52  0  l 

deviations,  and  if  any  deviation  were  chosen  54  0.1 

a*  random  there  would  be  only  one  chance  58  Q'Q 

in  ten  that  it  would  be  larger  than  the  cor-  60  0  0 
responding    z-value.      It    can    be    proved 


118  CRITERIA  OF  REJECTION 

mathematically  that  in  order  to  include  nine-tenths 
of  the  area  the  ordinate  must  be  drawn  2.44  times 
as  far  to  the  right  of  the  ?/-axis  as  the  line  which 
bisects  the  area  and  corresponds  to  the  dispersion. 

On  your  diagram  draw  the  ordinate  that  includes 
nine-tenths  of  the  area  and  make  sure  that  its  abscissa 
fulfills  the  condition  stated  above.  If  the  total  num- 
ber of  measurements  were  ten  how  many  would  most 
probably  be  represented  by  the  area  to  the  right  of  the 
ordinate?  How  many  if  the  number  of  measurements 
were  50?  How  many  if  the  number  were  4?  How 
many  if  6?  The  last  two  questions  should  be  answered 
to  the  nearest  whole  number. 

Since  the  ordinate  for  x  =  2.44  d  includes  nine-tenths 
of  the  area  and  the  limit  2.44  X  dispersion  includes 
nine-tenths  of  the  deviations  it  might  be  said  theoretic- 
ally and  rather  figuratively  that  if  there  were  only 
five  measurements  in  a  certain  series  the  number  of 
measurements  whose  deviations  were  greater  than 
this  limit  would -most  probably  be  just  one-half.  In 
other  words  the  limit  would  be  just  on  such  a  border 
line  that  if  it  was  decreased  we  should  expect  it  to 
exclude  one  measurement  rather  than  no  measure- 
ments, and  if  it  was  increased  we  should  expect  it  to 
exclude  no  measurements  rather  than  one  measurement. 

It  follows,  then,  that  no  one  of  a  series  of  five  meas- 
urements theoretically  ought  to  have  a  deviation  of 
more  than  2.44  times  the  dispersion.  This  being  the 
case  it  is  only  natural  to  consider  that  one  is  justified 
in  discarding  any  one  of  the  measurements  of  a  series 
of  five  if  its  deviation  does  exceed  this  limit.  Chauv- 
enet's  criterion  is  simply  an  extension  of  this  statement 


CRITERIA  OF  REJECTION 


119 


to  other  values  of  n  as  well  as  5.  The  column,  /,  of  the 
following  table  shows  the  limiting  value  of  x  for  which 
the  area  of  the  curve  y  =  e~x2  is  1  —  l/2n,  this  value 
being  expressed  in  terms  of  the  dispersion  or  probable 
error. 


n 

I 

log  I 

n 

I 

log  I 

n 

I 

log  I 

n 

I 

log  I 

1 
2 
3 
4 
5 

1.00 
1.71 
2.05 
2.27 
2.44 

000 
233 
312 
356 
387 

11 
12 
13 
14 
15 

2.97 
3.02 
3.07 
3.11 
3.15 

473 
480 

487 
493 
498 

21 
22 
23 
24 
25 

3.35 
3.38 
3.40 
3.43 
3.45 

525 
529 
532 
535 
538 

32 
34 
36 

38 
40 

3.59 
3.62 
3.65 
3.68 
3.70 

554 
556 
561 
566 
570 

6 

7 
8 
9 
10 

2.57 
2.67 
2.76 

2.84 
2.91 

410 
427 
441 
453 
464 

16 
17 
18 
19 
20 

3.19 
3.22 
3.26 
3.29 
3.32 

504 
508 
513 
517 
521 

26 
27 
28 
29 
30 

3.47 
3.49 
3.51 
3.53 
3.55 

540 
543 
546 
549 
551 

49 
64 
81 
100 
676 

3.81 
3.95 
4.06 
4.16 
5.00 

581 
597 
608 
619 
699 

Chauvenet's  criterion. 

It  should  be  carefully  kept  in  mind  when  consid- 
ering any  criterion  of  rejection  that  we  are  interested 
in  the  individual  measurements,  and,  accordingly,  the 
dispersion  to  which  the  criterion  applies  is  the  disper- 
sion of  the  individual  measurements,  not  the  dispersion 
of  the  average.  Chauvenet's  criterion  is  then  the  test 
of  whether  any  deviation  is  greater  than  I  times  the 
dispersion  of  the  individual  values  of  a  series  of  n 
measurements. 

The  Probable  Error. — It  will  be  noticed  that  the 
dispersion  has  also  been  called  the  " probable  error." 
By  this  time  the  student  ought  to  be  thoroughly  aware 
of  the  fact  that  the  dispersion  is  not  an  error  at  all, 
but  a  deviation.  If  he  also  realizes  that  deviations 
within  its  limits  are  no  more  probable  than  improbable 


120  CRITERIA  OF  REJECTION 

there  can  be  no  objection  to  his  using  the  term  that 
is  always  employed  by  physicists  in  speaking  of  this 
characteristic  deviation.  In  this  book  the  term  dis- 
persion has  been  used  in  order  to  avoid  repeatedly 
informing  the  student  that  it  is  an  error  and  repeatedly 
suggesting  that  there  is  something  very  probable  about 
it.  It  will  hereafter  be  spoken  of  as  the  probable  error, 
and  of  course  it  will  be  understood  that  it  is  used  in 
two  forms,  the  probable  error  of  the  individual  meas- 
urements and  the  probable  error  of  the  average. 

^In  an  experimental  determination  of  the  specific 
heat  of  lead  shot  the  following  values  were  obtained 
by  a  class  of  students.  Test  them  by  Chauvenet's 
criterion  to  determine  whether  any 'measurement  falls 

Q22  outside  of  the  theoretical  limits,  but  if  two  or 
.0309  more  such  values  are  found  reject  only  the  most 

032  divergent  one,  find  a'  new  average  for  those  that 
.0347  remain,  and  apply  the  criterion  to  them  in  turn. 
•  jjgg  Repeat  the  process,  if  necessary,  until  no  more 
.038  values  can  be  discarded,  and  then  state  the 

049      kes^  vame  obtainable  from  the  figures,  with  its 

''probable  error." 

Graphic  Approximation  to  Chauvenet's  Criterion. — 
Where  a  graphic  diagram  is  to  be  used  for  only  a  single 
series  of  numbers  instead  of  for  sets  of  values  of  two 
varying  quantities  it  is  advisable  to  use  a  horizontal 
scale  and  lay  off  the  individual  measurements  as  small 
dots  or  circles  unless  they  are  sufficiently  numerous  to 
allow  a  good  histogram  to  be  drawn. 

The  following  figures  are  the  experimental  values  of 
the  slope  of  the  first  "  black-thread "  diagram  as  ob- 
tained by  a  class  of  students:  .60,  .57,  .64,  .53,  .48, 


CRITERIA  OF  REJECTION  121 

.59,  .59,  .61.  Make  a  graphic  diagram  of  these  values, 
mark  the  median  and  quartiles,  and  lay  off  the  semi- 
interquartile  range  to  the  right  and  left  of  the  median 
as  many  times  as  is  indicated  by  Chauvenet's  criterion. 

Q1  ME    Q3 


i r 


t-*n — t — i — r 
50          54          58         62         66 

In  this  way  a  rough  application  of  the  criterion  can 
be  made  graphically  and  the  long  calculation  can  be 
avoided.     Determine  from  the  diagram  whether  any 
value  should  be  rejected  and  then  verify     ^  Qg6  .    , 
the   result  by  the  usual  form  of  calcu-     29  982     " 

lation  29-"°     " 

29.984     " 

Use  the  graphic  method  for  applying     29.984     " 
Chauvenet's  criterion  to  the  following  set     ^9 '  ggg 
of  barometer  readings :  29 . 977 

What  advantage  has  the  arithmetical     29  982 
method  over  the  graphic  method?  29.986 

Write     down,     in    your    own    words,     29  984 
just  what    it    is  that   is  represented  by 
(a)    the    probable    error   of    a    single    measurement, 
.6745\/Sd2/(n  —  1,  and  by   (6)jthe  probable  error  of 
the  average,  .6745\/Zd2/™O  —  1). 

Irregularities  of  Small  Groups. — The  probable  error, 
or  "  dispersion,"  cannot  be  considered  as  having 
much  meaning  in  cases  where  the  total  number  of 
measurements  is  less  than  ten,  and  even  with  ten 
"measurements  it  should  be  treated  with  a  certain 
amount  of  caution.  A  number  of  values  less  than  ten 
will  hardly  ever  give  a  histogram  of  their  frequency- 


122  CRITERIA  OF  REJECTION 

distribution  which  is  recognizably  similar  to  the  graph 
of  y  =  e~*2,  the  curve  which  all  unbiassed  measure- 
ments will  be  found  to  follow  if  they  are  sufficiently 
numerous. 

Justification  of  the  Criterion. — Likewise,  it  is  hardly 
worth  while  to  use  a  criterion  of  rejection  for  less 
than  ten  measurements.  The  example  given  above 
with  only  five  is  intended  merely  for  an  illustration 
of  the  method  of  using  the  criterion,  and  the  still 
smaller  values  in  the  table  are  only  of  theoretical 
importance.  Chauvenet's  criterion  is  not  to  be  con- 
sidered as  showing  that  any  one  measurement  is  a 
mistake,  but  merely  as  indicating  that  a  very  large 
deviation  is  such  a  rarity  that  it  would  have  an  unduly 
large  influence  upon  the  average  if  it  were  allowed  to 
remain  along  with  the  other  values  of  a  very  limited 
series  of  measurements. 

Wright's  Criterion. — Another  criterion  of  rejection, 
which  is  sometimes  employed,  is  that  of  Wright. 
According  to  this  the  arbitrary  rejection  of  a  single 
measurement  may  be  considered  if  its  deviation  is 
more  than  five  times  the  probable  error. 

Turn  back  to  the  graphic  diagram  of  the  table  on 
page  117,  and  notice  how  small  a  part  of  the  area  of 
the  curve  lies  to  the  right  of  the  ordinate,  x  =  50, 
representing  five  times  the  probable  error.  Turn  to 
the  table  of  values  for  Chauvenet's  criterion  and  note 
how  many  measurements  would  need  to  be  made  before 
"half  a  measurement"  would  be  likely  to  diverge  from 
the  average  five  times  as  far  as  the  probable  error. 

In  the  measurements  to  which  you  have  already 
applied  Chauvenet's  criterion  how  many  would  have 


LEAST  SQUARES  AND  VARIOUS  ERRORS       123 

been  rejected  if  Wright's  criterion  had  been  used 
instead?  What  are  the  relative  advantages  of  the  two 
criteria? 

Other  limiting  values,  which  give  practically  the 
same  result  as  Wright's  criterion,  are  four  times  the 
average  deviation,  and  three  times  the  standard  deviation. 

Comparison  of  Characteristic  Deviations. — Turn 
back  to  your  notes  on  the  use  of  logarithms  and  find 
the  graphic  diagram  of  y  =  e-*2.  Mark  off  the  following 
values  on  the  base  line,  p  =  .4769,  a  =  .5642,  and 
s  =  .7071.  These  represent  respectively  the  probable 
error,  the  average  deviation,  and  the  standard  devia- 
tion, and  are  roughly  proportional  to  10  :  12  :  15;  a 
better  approximation  may  be  found  with  the  aid  of 
the  slide  rule.  Draw  the  corresponding  ordinates,  and 
notice  that  the  last  one  meets  the  curve  at  the  point  of 
inflection,  that  is,  at  the  point  where  it  is  momentarily 
straight  as  it  changes  from  convex  upward  to  convex 
downward. 


XV.   LEAST  SQUARES  AND  VARIOUS 
ERRORS 

Apparatus. — Clock,  chronometer,  or  time  circuit, 
giving  audible  seconds;  watch  with  second-hand; 
slide  rule. 

The  Average  as  a  Least-Square  Magnitude. — The 
mathematical  principle  of  least  squares  is  that  when 
measurements  are  equally  trustworthy  their  best 
'representative  value  is  that  for  which  the  sum  of  the 
squares  of  the  deviations  has  the  lowest  numerical 
value.  It  is  upon  this  principle  that  the  use  of  the 


124       LEAST  SQUARES  AND  VARIOUS  ERRORS 

average  is  based,  for  it  is  easy  to  show  that  the  sum  of 
the  squares  of  the  deviations  of  any  particular  set  of 
numbers  (try  3,  3,  4,  5,  10)  will  be  greater  when  meas- 
ured from  some  other  value  (try  the  median)  than  when 
measured  from  the  average. 

Least  Squares  for  Conditioned  Measurements. — 
If  we  are  dealing  with  two  conditioned  measurements, 
as  in  the  case  of  the  x-  and  ^/-values  of  the  black- 
thread  experiment,  the  principle  of  least  squares  shows 
that  the  line  which  expresses  the  condition  or  gives 
the  law  of  relationship  between  the  two  variables 
must  be  so  placed  that  the  sum  of  the  squares  of  the 
distances  from  it  to  all  of  the  experimental  points  shall 
have  the  smallest  possible  value. 

The  x  and  y  of  any  one  of  the  points  cannot  in  general 
be  substituted  in  the  black-thread  equation,  y  =  a  + 
bx,  but  a  +  bx  —  y  will  have  some  small  positive  or 
negative  value  instead  of  being  equal  to  zero.  If  the 
various  points  are  considered  to  have  the  definite  posi- 
tions (xi,  t/i),  (x2j  2/2),  (x3j  2/3),  etc.,  it  will  be  found  that 
none  of  these  sets  of  values  will  satisfy  the  equation 
y  =  a  +  bx  or  a  +  bx  —  y  =  0,  but  will  give  such  a 
result  as  a  +  bxi  —  yi  =  di,  where  d  is  some  small 
quantity  whose  exact  value  need  not  be  determined; 
similarly,  the  other  points  will  give  a  +  bx%  —  y%  =  d-2, 
a  -\-  bx3  —  2/3  =  d$,  etc.,  and  according' to  the  principle 
of  least  squares  the  sum  diz  +  d^  +  d£  +  .  .  . ,  must 
be  as  small  as  possible.*  This  means  that  the  sum  of  the 
left-hand  members  of  the  equations,  or  2 (a  +  bxn—  yn)2 
must  have  its  minimum  value,  and  it  can  be  shown 

*  It  can  be  shown  mathematically  that  the  distance  from 
(x\y\)  to  y  =  a  +  bx  is  proportional  to  a  +  bxi  —  y\. 


LEAST  SQUARES  AND  VARIOUS  ERRORS       125 


by  processes  of  pure  mathematics  that  this  will  be  the 
case  if 

)  S  (xy)  -  Z  (y)  Z  (z2) 


_ 


and 


Z  (a;)  Z  (y)  -  nZ  (ay) 
~  (Zz)2  -  n  Z  (z2)      ' 


By  similar  processes  a,  6,  and  c  could  be  found  for 
the  equation  of  the  parabola  y  =  a  -\-  bx  +  ex2,  or  the 
appropriate  coefficients  for  curves  having  even  more 
complicated  equations,  but  the  processes  of  computa- 
tion becomes  so  tedious  that  it  is  better  to  replace  the 
variables,  as  explained  in  the  lesson  on  graphic  analysis, 
by  others  that  will  conform  to  the  straight  line  law. 

Tabulate  the  values  of  x  and 
y  for  the  black-thread  experiment 
on  page  70,  arranging  •  them  as 
shown  in  the  following  table,  but 
writing  the  proper  numerical 
values  in  the  spaces  marked  Sa; 
Sty,  2(xy),  and  2(>2).  Then  cal- 
culate the  values  of  a  and  b 
from  the  formula,  arranging  the 
work  neatly  and  being  careful  to 
avoid  using  the  wrong  algebraical 
signs  or  confusing  2  (x2)  with 
(Zx)2.  Keep  only  three  signifi- 
cant figures  in  the  final  results, 

Write  the  equation  represent- 

ing the  best  position  of  the  black  thread  in  the  form 
y  =  a  -f-  bx,  and  then  in  the  form  xja  +  y/b  =  1. 
Compare  the  calculated  values  of  the  intercepts  with 


X 

y 

xy 

X* 

1 

9.8 

9.8 

1 

2 

8.5 

17.0 

4 

3 

8.0 

24.0 

9 

4 

7.2 

28.8 

16 

5 

6.7 

33.5 

25 

6 

6.5 

39.0 

36 

7 

6.2 

43.4 

49 

8 

5.5 

44.0 

64 

9 

5.0 

45.0 

81 

10 

4.1 

41.0 

100 

11 

3.9 

42.9 

121 

12 

3.2 

38.4 

144 

13 

2.3 

29.9 

169 

Zy     z(zy)  |z(x2) 


126       LEAST  SQUARES  AND  VARIOUS  ERRORS 

your  experimental  values  obtained   in  the  lesson  on 
graphic  analysis. 

Probable  Errors  of  Indirect  Measurements. — All  the 
probable  errors  which  have  been  considered  previously 
have  been  probable  errors  of  direct  measurements. 
Suppose  an  indirect  measurement  is  to  be  obtained 
from  certain  direct  measurements  by  the  use  of  an 
appropriate  formula.  If  one  set  of  values  of  the 
direct  measurements  were  substituted  in  the  formula 
in  order  to  calculate  one  value  of  the  indirect  meas- 
urement, and  then  another  set  in  order  to  calculate  a 
second  value,  and  so  on,  it  would  be  an  easy  matter 
to  find  the  probable  error  of  the  resultant  set  of  indirect 
measurements  but  the  repeated  calculations  would  be 
much  more  laborious  and  time-consuming  than  the 
process  of  substituting  the  average  of  each  direct 
measurement.  Furthermore,  some  of  the  values 
needed  in  the  formula  may  be  predetermined  con- 
stants, such  as  those  mentioned  on  page  109,  which 
are  given  only  in  the  form  of  a  representative  mag- 
nitude and  its  probable  error.  The  question  then 
arises  as  to  the  way  in  which  the  probable  error  of  the 
indirect  measurement  is  influenced  by  the  size  of  the 
probable  errors  of  the  direct  measurements  on  which 
it  is  based. 

The  simplest  case  is  when  the  indirect  measurement 
is  merely  the  sum  of  the  two  independent  direct  meas- 
urements.    Let  the  direct  measurements  be 
a\  =±  pi         and         az  =t  pz, 

when  stated  in  the  form  of  average  and  probable  error, 
and  let  the  indirect  measurement  with  its  probable  error 

be 

A*P, 


LEAST  SQUARES  AND  VARIOUS  ERRORS       127 
so  that  A  =  «i  +  a2;   then  it  can  be  proved  that 

P2=    ?12+P22. 

Similarly  if  A  =  ai  —  a2  it  is  also  true  that  P2  is  equal 
to  7? i2  +  p22,  not  pi2  —  p22. 
If 

A  =  cioi, 

where  Ci  is  some  constant  not  subject  to  error,  it  can 
be  shown  that 

P2  =  Ci2pi2  or  P  =  cipi. 

In  general,  if 

A  =  ciai  ±  c2a2  =*=  c3a3  ±  .  .  . 
then 

P2    =    Cj2pi2   +  C22p22    +   C32ps2    +  •  •  • 

where  pn  is  the  probable  error  of  the  average  an. 

This  formula  can  be  used  to  find  the  probable  error 
of  an  algebraical  expression  when  the  probable  error  of 
each  of  its  terms  is  known. 

If  two  independent  measurements  are  multiplied 
together  the  probable  error  of  the  product  will  follow 
the  law  expressed  in  the  following  equation. 

If 

A  =  a\a<i 
then 

P2  =  pra22  +  ai2p22. 

Likewise  if 

A  = 

» 

then 

P2  = 


128      LEAST  SQUARES  AND  VARIOUS  ERRORS 

and  similarly  for  any  number  of  factors  ;  brut  it  is  more 
satisfactory  in  practice  to  make  use  of  the  relative 
probable  error  (relative  dispersion,  page  109)  as  in  the 
following  form,  which  is  easily  deducible  from  the  form 
just  given. 

If 

A  =ai  0,2  a3  .... 
then 

(P/A)2  =  (Pl/ai)2  +  (p2/a2)2  +  (p3/asr~  +  .  .  -  - 

A  =  ain 
then 

(P/A)2  =  (n2  plVai»)  or  P/A  =  np,/ai;   . 

and,  likewise,  if 

A  =  Ci  ain 
then 

P/A  =  npi/ai, 

the  constant  not  appearing  in  the  formula  if  the  rela- 
tive probable  error  is  used. 

In    general,   whether   the    exponents  are  positive, 
negative,  or  fractional,  if 

A  =  Ciaima->na3r.  .  .  . 
then 

(P/A)2  =  (mpi/aO*  +  (np2/a2)2  +  (rp3/a3)2  +  . 


This  formula  can  be  used  to  find  the  probable  error 
of  any  single  term  of  an  algebraical  expression  when 
the  probable  errors  of  its  factors  are  known. 

A  bowl  whose  interior  is  an  exact  segment  of  a  sphere 
is  found  to  have  a  depth  of  25.00  =*=  .02  centimetres 
and  a  diameter  across  the  top  of  50.00  =•=  .30  centi- 
metres. Find  its  capacity  from  the  formula  for  the 
volume  of  a  spherical  segment,  v  =  Trhr2/2  -f-  ?r/i3/6, 
where  h  is  the  height  or  depth  of  the  segment  and 
r  is  the  radius  of  its  circular  base;  find  the  probable 


LEAST  SQUARES  AND  VARIOUS  ERRORS       129- 

error  of  the  capacity  by  applying  the  second  general 
equation  to  each  term  of  the  formula  and  then  using 
the  first  equation  to  determine  the  final  result.  Notice 
the  relative  probable  error  of  the  radius,  r,  is  the  same 
as  that  of  the  diameter,  d.  Arrange  the  calculation 
systematically  in  order  to  avoid  numerical  mistakes, 
and  if  there  is  any  trouble  in  making  the  substitution 
write  out  each  step  of  the  process;  for  example: 

«!    =    25.          P2/A2  =  32(.02)2/252  log  25  =  1.3979 

Pl   =  .02  P2  =  32(.02)V2256/25262  log  7r2        0.9943 

m    =      3  =  7r2(.01)2254  4.0000 

ci    =  7T/6  5.5916 

A    =  7r^/6  2.5859 

log'P  = 

Notice  that  an  indirect  probable  error  which  depends 
upon  the  measurement  of  two  or  more  different  quan- 
tities always  assumes  the  form  \/x2  +  y2,  and  con- 
sequently will  be  more  decidedly  diminished  by  reduc- 
ing the  larger  of  the  two  independent  probable  errors 
than  by  attempting  to  improve  the  more  accurate 
measurement.  Show  that  \/52  -f  22  is  reduced  by 
41%  if  the  5  is  changed  to  2.5,  but  only  by  5%  if  the  2 
is  changed  to  1. 

Systematic  Errors. — It  has  been  shown  that  errors 
may  be  either  accidental  or  constant.  There  is  another 
class  of  errors,  often  included  under  the  term  constant 
errors,  in  which  the  error  is  not  actually  constant,  nor 
does  it  vary  according  to  the  law  of  probability.  This 
is  the  class  of  systematic  errors,  or  errors  that  undergo 
"^  more  or  less  regular  change  during  the  course  of 
making  a  set  of  measurements.  They  may  be  sub- 
divided into  progressive  errors,  which  show  a  steady 
9 


130      LEAST  SQUARES  AND  VARIOUS  ERRORS 

increase  (or  decrease)  from  one  determination  to  the 
next,  and  periodic  errors  which  increase  for  a  number  of 
measurements,  then  decrease,  and  then  repeat  the 
previous  cycle  or  period.  Where  systematic  errors  are 
absent  a  comparison  of  any  measurement  of  a  series 
with  the  preceding  one  will  tend  to  show  an  increase  in 
the  numerical  value  about  as  often  as  a  decrease;  a 
fact  that  can  easily  be  tested  by  writing  between 
each  two  successive  values  a  plus  sign,  a  minus  sign, 
or  a  zero,  according  as  the  second  value  is  greater 
than,  less  than,  or  equal  to,  the  first,  and  then  com- 
paring the  number  of  the  plus  signs  with  that  of  the 
minus  signs.  Where  progressive  errors  have  been 
greater  than  accidental  errors  there  may  be  all  plus 
signs  or  all  minus  signs  as  the  result  of  applying  the 
test;  and  if  the  systematic  errors  are  periodic  there 
will  be  alternate  groups  of  plus  signs  and  minus  signs. 

In  an  experiment  in  which  water  in  a  reservoir  was 
drawn  up  into  a  tube  by  suction  and  successive  read- 
ings of  its  height  were  made  values  having  the  fol- 
lowing decimals  were  obtained  in  order:  .76,  .74,  .70, 
.62,  .63,  .61,  .55,  .56,  .51,  .50,  .44,  .44,  .39,  .40,  .35,  .35. 
Are  the  results  probably  affected  by  progressive  errors, 
or  periodic  errors,  or  neither?  Why? 

Stand  where  you  can  hear  the  clock  beat  seconds  and 
read  the  time  indicated  by  your  watch.  Every  seven 
seconds  as  indicated  by  the  clock  read  the  seconds  and 
estimated  tenths  of  a  second  from  the  watch  and  state 
the  result  to  another  student,  who  will  take  down  the 
values  in  his  note-book.  After  three  or  four  minutes 
change  places  with  him  and  note  down  the  time  as  he 
reads  it  off.  Everv  seven-second  interval  should  have 


LEAST  SQUARES  AND  VARIOUS  ERRORS      131 

its  time  by  the  watch  noted,  for  a  full  period  of  seven 
minutes. 

See  that  you  have  the  complete  table  of  sixty  values 
in  your  own  note-book,  and  mark  the  observed  tenths  of 
a  second  with  a  plus  sign  where  they  increase  from  one 
observation  to 'the  next  and  with  a  minus  sign  where 
they  decrease.  With  most  watches  it  will  be  found 
that  the  second  hand  is  not  pivoted  in  the  exact  centre 
of  the  graduated  circle  and  the  periodic  error  will  be 
shown  very  distinctly. 

hr  min  sec  hr  min  sec  hr  min  sec 


4:37:65.2  , 

4:40:25.8 

4:42:45.2 

38:12.31 
19.6+ 
26.  6U 

32.6" 
39.  3~ 
46.1 

52.07 

59.1  + 
43:06.3+ 

33.5 
40.3 
47.17! 

53.0, 
60.2+ 
41:07.4+ 

13.5  + 
20.6+ 

27.8+ 

54.lg 
39:01.  IT 

08.21 

14.6+ 

21.7  + 
28.  7U 

34.6 
41.3 
48.1 

15.4+ 

35.6 

55.0  , 

22.6+ 
29.  6U 
36.4 
43.1 

42.  2~ 
49.  10 
56.1? 

42:03.2+ 

44:02.3  + 
09.4+ 
16.6+ 

23.8  + 

50.  0~ 

10.3  + 

30.7 

57.  02 
64.2  + 

12.5  + 

24.7  + 

37.4 
44.1,, 

40:11.4+ 

31.6 

51.  1° 

18.  6T 

38.  4~ 

58.  1° 

+ 

4:45:05.3+ 

Draw  a  graphic  diagram  in  which  the  abscissae  repre- 
sent the  integral  part  of  the  number  of  seconds  in  the 
table,  and  the  ordinates  represent  the  corresponding 
"tenths  of  a  second.  Draw  a  smooth  curve  to  eliminate 
accidental  errors  in  the  determination  of  time. 

Summarize  the  result  by  stating  "the  pivot  of  the 


132      LEAST  SQUARES  AND  VARIOUS  ERRORS 


second  hand  of  the  watch  is  displaced  toward  the 
figure ...  of  the  dial  by  an  amount  equal  to  the  length 
of . . .  seconds'  divisions  on  the  graduated  circle." 


•v.. 


40 


50 


10 


Explain  how  the  periodic 
error  can  be  eliminated  in 
case  such  a  watch  is  used  for 
determining  intervals  of  time. 

The  list  of  figures  given 
on  page  130  was  obtained 
from  a  determination  of 
specific  gravity  by  Hare's 
method.  If  the  lower  ends 
of  two  upright  tubes  dip  into 
two  separate  reservoirs  while 
their  upper  ends  are  both 
joined  to  a  third  tube  from 
which  the  air  can  be  partially 
exhausted  it  can  easily  be 


LEAST  SQUARES  AND  VARIOUS  ERRORS      133 


pure 
water 


salt 
solution 


proved  that  the  heights  to  which  the  fluids  are  raised 
will  be  inversely  proportional  to  their  densities;  so  that 
if  a  fluid  whose  density  is  unity  is  raised  to  a  height  hi, 
and  a  heavier  fluid  to  a  lesser  height 
h 2,  the  density  or  specific  gravity  of 
the  latter  must  be  hi/hz.    The  com- 
plete list  of  determinations  of  height 
included  readings  of  both  columns  of 
liquid;    they  were   made    at  approxi- 
mately equal   intervals   of   time,   and 
in  the  order  in  which  they  are  given 
in  the  table. 

If  the  density  is  calculated  by 
dividing  75.76  by  73.06  it  is  evident 
that  the  resulting  figure  will  be  too 
large,  for  the  height  of  the  water 
had  fallen  somewhat  below  75.76 
when  the  reading  of  the  salt  solution, 
73.06,  was  taken;  and  if  75.74  is 
divided  by  73.06  the  result  will  be 
too  small,  for  the  salt  solution  did  not 
stay  at  73.06  while  the  reading  75.74 
was  being  taken.  Obviously  the  aver- 
age of  75.76  and  75.74  must  be 
divided  by  73.06,  or  75.74  must  be 
divided  by  the  average  of  73.06  and 
73.04,  or  some  other  combination 
used  in  which  the  average  height  of 
one  column  of  liquid  must  have  oc- 
4drred  at  the  same  time  as  the  average  height  of  the 
other.  This  method  of  eliminating  progressive  errors 
is  used  in  the  process  of  weighing  with  a  delicate  bal- 


75.76 
75.74 
75.70 

75.62 

.63 
.61 


,55 
.56 


.51 
.50 


44 
44 


.39 
.40 


.35 
.35 


73.06 
73.04 

73.00 

72.98 

.95 

.94 


89 

88 


84 
84 


79 

,78 


75 
70 


.77 
.77 


134      LEAST  SQUARES  AND  VARIOUS  ERRORS 

ance  and  in  many  other  processes  of  physical  measure- 
ment. 

What  set  of  values  near  the  end  of  the  table  can  be 
used  in  the  same  way?  Make  five  different  calculations 
of  density  from  successive  parts  of  the  table  and  see 
whether  they  show  any  evidence  of  progressive  error. 

Constant  Errors. — It  has  already  been  stated  that 
constant  errors  are  more  troublesome  than  accidental 
errors  and  that  the  latter  give  very  little  aid  in  deter- 
mining the  former.  It  is  not  the  target  (page  86)  that 
is  found  from  individual  measurements  but  only  the 
centre  of  clustering,  and  characteristic  deviations  show 
only  how  close  determinations  come  to  one  another, 
not  how  close  they  come  to  the  truth. 

Some  constant  errors  are  easily  corrected  with  the 
aid  of  theoretical  considerations;  others  may  be  very 
difficult  to  eliminate.  Unfortunately  there  is  no  in- 
fallible rule  for  detecting  them,  and  each  experimental 
problem  has  its  own  special  sources  of  error.  The  two 
beam-arms  of  a  balance  may  be  unequal,  so  that  all 
weighings  are  proportionately  erroneous;  the  end  of 
a  metre-stick  may  be  worn,  so  that  every  setting  of 
the  zero-point  is  inaccurate;  the  neutral  tint  of  litmus 
may  be  faultily  judged,  so  that  a  chemical  determina- 
tion is  biassed.  Consider  such  a  simple  process  as  the 
determination  of  atmospheric  pressure  with  a  mer- 
curial barometer.  The  vacuum  at  the  top  is  never 
perfect  and  there  is  often  capillary  action,  both  making 
the  reading  too  low.  If  the  barometer  and  its  attached 
scale  do  not  hang  vertically  every  apparent  reading 
will  be  too  high.  The  scale  itself  is  too  long  or  too 
short  except  at  a  single  temperature,  and  the  mercury 


LEAST  SQUARES  AND  VARIOUS  ERRORS      135 

may  have  its  accepted  standard  density  only  at  a  dif- 
ferent temperature  from  the  one  that  it  has  when  the 
observation  is  made.  Even  if  its  density  is  standard 
the  height  of  a  column  that  will  give  a  definite  pressure 
will  depend  upon  the  strength  of  gravitational  attrac- 
tion and  this  varies  with  the  latitude  and  altitude  of 
the  instrument.  If  an  aneroid  barometer  is  to  be  used 
instead  of  a  mercurial  one  its  mechanism  introduces 
still  more  sources  of  error. 

It  is  evident  that  the  amount  of  constant  error  can 
be  varied  by  changing  observers,  apparatus,  methods, 
and  times  of  observation;  and  the  more  radically 
different  the  sets  of  conditions  are  made  the  better,  in 
all  probability,  will  be  the  mutual  neutralization  of 
constant  errors  when  the  weighted  average  is  taken. 
In  practice,  the  values  for  most  of  the  constants  of 
nature  have  been  obtained  under  such  varying  con- 
ditions. Atomic  weights  are  obtained  from  various 
inter-relations  of  chemical  compounds  obtained  from 
different  sources  and  by  different  methods.  The 
surface  tension  of  water  may  be  measured  by  the  hang- 
ing drop  method,  by  the  capillary  wave  method,  by 
the  vibrating  jet  method,  etc.  The  size  of  the  molecules 
of  a  gas  may  be  calculated  from  the  rate  at  which  heat 
is  conducted  through  them,  from  the  "covolume  con- 
stant," b,  of  Van  der  Waal's  equation,  from  experimental 
determinations  of  the  viscosity  of  the  gas,  from  meas- 
urements of  the  maximum  density  obtainable  by  cooling 
and  liquefying  or  solidifying  it,  etc.  If  various  deter- 
minations agree  closely  in  spite  of  the  employment  of 
essentially  different  methods  it  becomes  more  probable 
that  constant  errors  have  been  satisfactorily  removed, 


136      LEAST  SQUARES  AND  VARIOUS  ERRORS 

but  it  can  never  be  certain  that  all  of  these  methods 
have  not  some  common  source  of  error  which  would 
be  eliminated  only  by  using  some  entirely  different 
method.  Constant  watchfulness,  as  stated  on  page  87, 
and  the  exercise  of  good  judgment  are  of  the  greatest 
importance  in  guarding  against  constant  errors. 
In  the  student's  future  laboratory  work  various 
" sources  of  error"  that  have  been  found  by  previous 
experimenters  will  be  explicitly  stated.  Many  of 
them  will  be  sources  of  constant  error,  and  both  his 
natural  ability  and  his  progress  in  learning  will  be 
tested  by  his  treatment  of  them. 


TABLES 


EXPLANATORY  NOTE 

The  use  of  the  table  of  squares  will  be  self-evident. 
Notice  that  the  square  of  a  number  between  100  and 
110,  say  of  107,  consists  of  five  figures  which  are,  in 
order,  1,  2n,  n2,  or  1,  14,  49.  The  square  of  any  number 
between  100  and  200  can  be  found  by  the  same  pro- 
cess, " carrying"  mentally.  Thus 

1122  =  1  1732  =  1 

24  146 

144  5329 


12544  29929 

If  either  Z(d2)/n  or  Z(d2)/n(n  -  1)  is  located  be- 
tween two  consecutive  numbers  in  the  third  column, 
(n  =«=  l/2)/(.67449)2,  of  the  samejable^jthen  the  value 
of  .6745\/2(d2)/n  or  .6745\/Z(d2)M  (n  -  1),  as  the  case 
may  be,  will  be  found  opposite  it  in  the  first  column. 
A  very  rough  mental  calculation  will  prevent  taking  a 
value  which  is  V  10  times  too  large  or  small. 

The  table  of  circular  functions  gives  the  "  radian 
value,"  natural  sine,  cosine,  tangent,  and  cotangent 
for  every  degree  of  the  quadrant,  also  the  logarithmic 
sine  and  cosine.  By  subtracting  the  two  latter  from 
each  other  and  from  zero  any  of  the  six  logarithmic 
functions  may  be  obtained  from  the  table  by  inspection. 
"Sines  and  cosines  of  any  intermediate  values  can  safely 
be  obtained  by  interpolation,  and  tangents  up  to  tan 
70°.  For  the  sine,  tangent,  and  numerical  measure  of 


140  EXPLANATORY  NOTE 

a  small  angle  the  equations  at  the  corners  of  the  table 
should  be  used. 

In  the  four-place  logarithm  table,  as  well  as  in  the 
five-place  table,  the  approximate  tabular  difference  has 
been  given  at  such  frequent  intervals  as  to  make  it 
unnecessary  to  subtract  anything  except  the  final 
digit.  In  the  five-place  table  proportional  parts  have 
been  omitted,  as  they  cannot  be  made  trustworthy  in 
such  a  table  on  two  pages.  Interpolations  may  be 
calculated  on  paper;  or,  after  a  little  practice,  can 
easily  be  worked  out  mentally  by  using  three-figure 
logarithms  taken  from  the  table,  and  adding  them  from 
left  to  right  in  order  to  calculate  products.  To  inter- 
polate an  antilogarithm  mentally,  proceed  as  follows: 
Using  three  decimal  places  take  out  first  the  cologar- 
ithm  of  the  tabular  difference;  add  it  to  the  logarithm 
of  the  tabular  excess1  and  the  antilogarithm  of  the 
sum  will  give  the  fourth  and  fifth  figures  of  the  required 
number. 

When  three-figure  logarithms  are  wanted  they  should 
be  taken  from  the  five-place  table.  On  the  first  page 
or  in  the  first  column  of  the  second  page  a  final  50 
or  500  is  simply  to  be  dropped  (as  indicated  by  the 
minus  sign  after  it);  elsewhere  the  preceding  figure  is 
to  be  increased  by  unity. 

1  I.e.,  the  difference  between  the  given  logarithm  and  the  next 
lower  tabular  value. 


SQUARES  AND  CIRCULAR  FUNCTIONS 


141 


(n±f)2 

Cn±i)2 

i 
RAD  DEG  TAN  SIN  ^/jy  ^5  COS  COT  ^IwSSS* 

*  .674492 

"  .674492 

10  1OO  21762 

60  3600  77819 

0000   0  :  0000  0000  -oo   0    1   oo  ;  90  7T/2 

JU  94904 

11  121  24234 
12  144  j£™£ 

uu  ovH/u  ,,/*  i  -  /. 

A1    *27O1  WnfcvD 

Si  3J21  83i38 

0^   00*14  fi^QRd. 

0175   1  0175  0175  2419  9999  9998  5729  |  89  1553 
0349   2  0349  0349  5428  9997  9994  2864  •  88  1536 

13   169  4rjnpi 

14  196  4Q915 

1  11  S 

0524   3  !  0524  0523  7188  9994  9986  1908  87  1518 
0698   4  0699  0698  8436  9989  9976  1430  '86  1501 

17  289  59843 

66  4356  J§304  j 
67  4489  i//,Y- 

0873   5  0875  0872  9403  9983  9962  1143 

85  1484 

18  324  7523() 

fQ  4781  1°314 

1047   6  1051  1045  0192  9976  9945  9514 

84  1466  I 

19  361  | 

10617 

1222   7  1228  1219  0859  9968  9925  8144 

83  1449 

1396   8  1  1405  1392  143(5  995S  9903  7115 

82  1431 

20  400  Q0.,— 

70  4900 

1571   9  !  1584  1564  1943  9946  9877  6314 

81  1414 

21  441  Vm«i 

71  5041  1  007 

22  484     , 

72  5184  }}{£( 

1745  10 

1763  1736  2397  9934  9848  5671 

80  1396 

23  529  {I}2,8. 

73  5329  ,'~- 

24  576    "• 

74  5476  }iSx 

1920  1  1 

1944  1908  2806  9919  9816  5145 

79  1379 

25  625  142qo 

75  5625  {gig 

2094  12 

2126  2079  3179  9904  9781  4705 

78  1361 

26  676  }tj93 

/O   o//O  i  ooft-4 

2269  13 

2309  2250  3521  9887  9744  4331 

77  1344 

27  729  jg*23 

77  5929  J0900 

2443  14 

2493  2419  3837  9869  9703  4011 

76  1326 

28  784  170=4 

78  6084  JXfJff 

29  841  1'8'" 

79  6241  l] 

2618  15 

2679  2588  4130  9849  9659  3732 

75  1309 

.  19129 

13893 

30   900  on 

80  6400 

2793  16 

2867  2756  4403  9828  9613  3487 

74  1292 

31  961  20148 

81  6561  H«An 

2967  17 

3057  2924  4659  9806  9563  3271 

73  1274 

32  1024  ooojo 

82  6724  it,!.. 

3142  18 

3249  3090  4900  9782  9511  3078 

72  1257 

33  1089  Sffia 

83  6889  J^g 

3316  19 

3443  3256  5126  9757  9455  2904 

71  1239 

34  1156  Sjrjg 
35  1225  257[J2 

or  ^'l-)  15695 

3491  20 

3640  3420  5341  9730  9397  2747 

70  1222 

3§  }296  29284 

86  7396  ,  r  Atn 
87  7569  10**'*' 

3665  21 

3839  3584  5543  9702  9336  2605 

69  1204 

38  1444  3091} 

Of     1  '  JUJ7   1^1^9(1 

88  7744  ,-.', 

3840  22  4040  3746  5736  9672  9272  2475  68  1187 

39  1521  d-°M 

89  7921  1721b 

4014  23 

4245  3907  5919  9(540  9205  2356  67  1169 

34296 

17607 

4189  24 

4452  4067  6093  9607  9135  2246  66  1152 

40  1600  360.4 
41  1681  22»« 

90  8100  1C 

91  8281  ™ 

4363  25 

4663  4226  6259  9573  9063  2145 

65  1134 

42  17(54  3g'7Q3 

92  8464 

4538  26 

4877  4384  6418  9537  8988  2050 

64  1117 

43  1849  4.1594. 

93  8649  10917 

4712  27 

5095  4540  6570  9499  8910  1963 

63  1100 

44  1936  yioi-oo 
4"  9fi9r  43o28 

94  8836  iqpoQ 

4887  28 

5317  4695  6716  9459  8829  1881 

62  1082 

4fi  911  «  45506 

4O  Z11O  47=90 

95  9025  ofio47 
96  9216  ^XY^o 

50(51  29 

5543  4848  6856  9418  8746  1804 

61  1065 

47  2209  4Q—V 
48  2304  rV7n- 

97  9409  2Q8gt] 

5236  30 

5774  5000  6990  9375  8660  1732 

60  1047 

49  24°'  MS59 

*  *""  2i?S 

5411  31 
5585  32 

6009  5150  7118  9331  8572  1664 
6249  5299  7242  9284  8480  1600 

59  1030 
58  1012 

50  2500  rm-7 

100  10000  999n, 

5760  33  6494  5446  7361  9236  8387  1540 

57  9948 

51  2601  ° 

101  10201  9^,' 

5934  34  6745  5592  7476  9186  8290  1483 

56  9774 

52  2704  5J?§? 

102  10404  r,.  04 

53  2809  2EH£ 

103  10609  0^47 

6109  35 

7002  5736  7586  9134  8192  1428 

55  9599 

54  2916  °|91& 

104  10816  94^4 

55  3025  fi7707 

105  11025  044,'- 

6283  36 

7265  5878  7692  9080  8090  1376 

54  9425 

56  3136  2   . 

106  1123(5  Xlooo  • 

6458  37 

7536  6018  7795  9023  798(5  1327 

53  9250 

57  3249  IRi^r 

107  11449  9=47,9 

6632  38 

7813  6157  7893  89(55  7880  1280 

52  9076 

58  3364  7=99= 

108  mm  o-^r 

6807  39 

8098  6293  7989  8905  7771  1235 

51  8901 

59  3481  /0^-° 

109  11881 

77819 

26356  i 

6981  40 

8391  6428  8081  8843  7660  1192 

50  8727 

7156  41 

8693  6561  8169  8778  7547  1150 

49  8552 

7330  42 

9004  6691  8255  8711  7431  1111 

48  8378 

SYMBOL,   CONS 

TANT   LOGARITHM 

7505  43 

9325  6820  8338  8641  7314  1072 

47  8203 

TT        3.141593  0.4971499 

7679  44 

9(157  6947  8418  8569  7193  1036 

46  8029 

Zl   lSO°/7r=«. 

=57° 

!295784'J  1.758  1226    7S")4  4r>    !  7071  S405  SH)r>  7071  1000 

45  7854 

=  3437'.747  3.5362739 

=  206264" 

.80625  5.3144251 

"182S2  0.4342945    '  =,1').'i.X  COT  COS  ,  *('  7:V,r  SIN  TAN  DEC!  HAD 

'  M       0.4342943  9.6377843  i 

i.o-toi-i  ,        ui/.a  am 

142 


FOUR-PLACE  LOGARITHMS 


0 

1234 

5 

6789 

D 

PP 

10 

0000 

0043  0086  0128  0170 

0212 

0253  0294  0334  0374 

40 

1 

43 

42 

41 

40 

11  0414  !  0453  0492  0531  0569  0607 

0645  0682  0719  0755  37 

12  0792  i  0828  0864  0899  0934  0969  1004  1038  1072  1106  33 

1 

4.3 

4.2 

4.1 

4.0 

13 

1139  1173  1206  1239  1271  1303  !  1335  1367  1399  1430  31  2 

8.6 

8.4  ! 

8.2 

8.0 

14 

1461 

1492  1523  1553  1584  1614  1644  1673  1703  1732  29  3 

12.9 

12.6 

12.3 

12.0 

4 

17.2 

16.8 

16.4 

16.0 

15 

1761 

1790  1818  1847  1875  1903  !  1931  1959  1987  2014 

27  5 

21.5 

21.0 

20.5 

20.0 

6 

25.8 

25.2 

24.6 

24.0 

10 

2041 

2068  2095  2122  2148  !  2175  2201  2227  2253  2279  25  7 

30.1 

29.4 

28.7 

28.0 

17 

2304 

2330  2355  2380  2405  2430  2455  24SO  2504  2529  24  S 

34.4 

33.6 

32.8 

32.0 

18 

2553 

2577  2601  2625  2648  2672  '  2695  2718  2742  2765  i  23 

9 

38.7 

37.8 

36.9 

36.0 

19 

2788 

2810  2833  2856  2878  2900  2923  2945  2967  2989 

21 

20 

3010 

3032  3054  3075  3096  3118  3139  3160  3181  3201 

21 

39 

38 

37 

36 

21 

3222  3243  3263  3284  3304  !  3324  3345  3365  3385  3404 

20 

1 

3.9 

3.8 

3.7 

3.6 

22 

3424 

3444  3464  3483  3502  3522  3541  35(50  3579  3598 

19  2 

7.8 

7.6 

7.4 

7.2 

23 

3617  3636  3655  3674  3692  3711  3729  3747  3766  3784  18  3 

11.7 

11.4 

11.1 

10.8 

24 

3802 

3820  3838  3856  3874 

3892  3909  3927  3945  3962 

17  14 

15.6 

15.2 

14.8 

14.4 

15 

19.5 

19.0 

18.5 

18.0 

25 

3979 

3997  4014  4031  4048 

4065  4082  4099  4116  4133 

17 

<i 

23.4 

22.8 

22.2 

21.6 

7 

27.3 

26.6 

25.9 

25.2 

26 

4150 

4166  4183  4200  4216  4232 

4249  4265  4281  4298 

16 

8 

31.2 

30.4 

29.6 

28.8 

27 

4314 

4330  4346  4362  4378  !  4393  4409  4425  4440  4456 

16 

9 

35.1 

34.2 

33.3 

32.4 

28 

4472 

4487  4502  4518  4533  4548  4564  4579  4594  4609  !  15 

29 

4624 

4639  4654  4669  4683  4698 

4713  4728  4742  4757  !  14 

35 

34 

33 

32 

30 

4771 

4786  4800  4814  4829 

4843 

4857  4871  4886  4900 

14 

1 

3.5 

3.4 

3.3 

3.2 

31 

4914 

4928  4942  4955  4969  4983  4997  5011  5024  5038  i  13  2   7.0 

6.8 

6.6 

6.4 

32 

5051 

5065  5079  5092  5105  1  5119  1  5132  5145  5159  5172 

13  3  10.5 

10.2 

9.9 

9.6 

33 

5185 

5198  5211  5224  5237 

5250  5263  5276  5289  5302 

13  4  14.0 

13.6 

13.2 

12.8 

34 

5315 

5328  5340  5353  5366 

5378 

5391  5403  5416  5428 

13 

5 

17.5 

17.0 

16.5 

16.0 

(i 

21.0 

20.4 

19.8 

19.2 

35 

5441 

5453  5465  5478  5490 

5502 

5514  5527  5539  5551 

12  7 

24.5 

23.8 

23.1 

22.4 

8 

28.0 

27.2 

26.4 

25  .  (j 

36 

5563 

5575  5587  5599  5611 

5623 

5635  5647  5658  5670 

12 

9 

31.5 

30.6 

29.7 

28.8 

37 

5682 

5694  5705  5717  5729 

5740 

5752  5763  5775  5786 

12 

38 

5798  5809  5821  5832  5843 

5855 

5866  5877  5888  5899  12 

39 

5911 

5922  5933  5944  5955 

5966 

5977  5988  5999  6010 

11 

31 

30 

29 

28 

40 

6021 

6031  6042  6053  6064 

6075 

6085  6096  6107  6117 

11 

1   3.1 

3.0 

2.9 

2.8 

2 

6.2 

6.0 

5.8 

5.6 

41 

6128 

6138  6149  6160  6170  6180 

6191  6201  6212  6222 

10  3   9.3 

9.0 

8.7 

8.4 

42 

6232 

6243  6253  6263  6274  6284  6294  6304  6314  6325  10  4  12.4 

12.0 

11.6 

i  11.2 

43 

6335 

6345  6355  6365  6375  6385  1  6395  6405  6415  6425  10  5  '  15.5 

15.0 

14.5 

14.0 

44 

6435 

6444  6454  6464  6474  6484 

6493  6503  6513  6522 

10 

6  18.6 

18.0 

17.4 

I  16.8 

7 

21.7 

21.0 

20.3 

19.6 

45 

6532 

6542  6551  6561  6571 

6580 

6590  6599  6609  6618 

10 

8 

24.8 

24.0 

23.2 

!  22.4 

9 

27.9 

27.0 

26.1 

25.2 

46 

6628 

6637  6646  6656  6665 

6675 

6684  6693  6702  6712 

9 

47 

6721 

6730  6739  6749  6758 

6767 

6776  6785  6794  6803 

9 

48 

6812 

6821  6830  6839  6848 

6857 

6866  6875  6884  6893 

9 

!  27 

26 

25 

24 

49 

6902 

6911  6920  6928  6937 

6946 

6955  6964  6972  6981 

9 

1 

2.7 

2.6 

2.5 

2.4 

50 

6990 

6998  7007  7016  7024 

7033 

7042  7050  7059  7067 

9 

2 

5.4 

5.2 

5.0 

1  4.8 

3 

8.1 

7.8 

7.5 

7.2 

51 

7076 

7084  7093  7101  7110 

7118 

7126  7135  7143  7152 

8 

4 

10.8 

10.4 

10.0 

9.6 

52 

7160 

7168  7177  7185  7193 

7202  !  7210  7218  7226  7235 

8  5  13.5 

13.0 

12.5 

12.0 

53 

7243 

7251  7259  7267  7275 

7284  1  7292  7300  7308  7316 

8 

6  16.2 

15.6 

15.0 

14.4 

54 

7324 

7332  7340  7348  7356 

7364  7372  7380  7388  7396 

8 

7  18.9 

18.2 

17.5 

16.8 

8 

21.6 

20.8 

20.0 

!  19.2 

55 

7404 

7412  7419  7427  7435 

7443 

7451  7459  7466  7474 

8 

9 

24.3 

23.4 

22.5 

21.6 

i 

FOUR-PLACE  LOGARITHMS 


143 


0 

1234 

5 

6789 

D 

P  P 

55 

7404 

7412  7419  7427  7435 

7443 

7451  7459  7466  7474 

8 

23 

22 

21 

20 

56 

7482 

7490  7497  7505  7513 

7520 

7528  7536  7543  7551 

57 

7559 

7566  7574  7582  7589 

7597 

7604  7612  7619  7627 

1 

2.3 

2.2 

2.1 

2.0 

58 

7634 

7642  7649  7657  7664 

7672 

7679  7686  7694  7701 

2 

4.6 

4.4 

4.2 

4.0 

59 

7709 

7716  7723  7731  7738 

7745 

7752  7760  7767  7774 

3 

6.9 

6.6 

6.3 

6.0 

4 

9.2 

8.8 

8.4 

8.0 

60 

7782 

7789  7796  7803  7810 

7818 

7825  7832  7839  7846 

7 

5 

11.5 

11.0 

10.5 

10.0 

8 

13.8 

13.2 

12.6 

12.0 

61 

7853 

7860  7868  7875  7882 

7889 

7896  7903  7910  7917 

7 

16.1 

15.4 

14.7 

14.0 

62 

7924 

7931  7938  7945  7952 

7959 

7966  7973  7980  7987 

8 

18.4 

17.6 

16.8 

16.0 

63 

7993 

8000  8007  8014  8021 

8028 

8035  8041  8048  8055 

9 

20.7 

19.8 

18.9 

18.0 

64 

8062 

8069  8075  8082  8089 

8096 

8102  8109  8116  8122 

65 

8129 

8136  8142  8149  8156 

8162 

8169  8176  8182  8189 

6 

19 

18 

17 

16 

66 

8195 

8202  8209  S215  8222 

8228 

8235  8241  8248  8254 

1 

1.9 

1.8 

1.7 

1.6 

67 

8261 

8267  8274  8280  8287 

8293 

8299  8306  8312  8319 

2 

3.8 

3.6 

3.4 

3.2 

68 

8325 

8331  8338  8344  8351 

8357 

8363  8370  8376  8382 

3 

5.7 

5.4 

5.1 

4.8 

69 

8388 

8395  8401  8407  8414 

8420 

8426  8432  8439  8445 

4 

7.6 

7.2 

6.8 

6.4 

5 

9.5 

9.0 

8.5 

8.0 

70 

8451 

8457  8463  8470  8476 

8482 

8488  8494  8500  8506 

7 

(i 

11.4 

10.8 

10.2 

9.6 

7 

13.3 

12.6 

11.9 

11.2 

71 

8513 

8519  8525  8531  8537 

8543 

8549  8555  8561  8567 

8 

15.2 

14.4 

13.6 

12.8 

72 

8573 

8579  8585  8591  8597 

8603 

8609  8615  8621  8627 

9 

17.1 

16.2 

15.3 

14.4 

73 

8633 

8639  8645  8651  8657 

8663 

8669  8675  8681  8686 

74 

8692 

8698  8704  8710  8716 

8722 

8727  8733  8739  8745 

15 

14 

13 

12 

75 

8751 

8756  8762  8768  8774 

8779 

8785  8791  8797  8802 

(i 

1 

1.5 

1.4 

1.3 

1.2 

76 

8808 

8814  8820  8825  8831 

8837 

8842  8848  8854  8859 

2 

3.0 

2.8 

2.6 

2.4 

77 

8865 

8871  8876  8882  8887 

8893 

8899  8904  8910  8915 

3 

4.5 

4.2 

3.9 

3.6 

78 

8921 

8927  8932  8938  8943 

8949 

8954  8960  8965  8971 

4 

6.0 

5.6 

5.2 

4.8 

79 

8976 

8982  8987  8993  8998 

9004 

9009  9015  9020  9025 

5 

7.5 

7.0 

6.5 

6.0 

(i 

9.0 

8.4 

7.8 

7.2 

80 

9031 

9036  9042  9047  9053 

9058 

9063  9069  9074  9079 

6 

7 

10.5 

9.8 

9.1 

8.4 

8 

12.0 

11.2 

10.4 

9.6 

81 

9085 

9090  9096  9101  9106 

9112 

9117  9122  9128  9133 

9 

13.5 

12.6 

11.7 

10.8 

82 

9138 

9143  9149  9154  9159 

9165 

9170  9175  9180  9186 

83 

9191 

9196  9201  9206  9212 

9217 

9222  9227  9232  9238 

84 

9243 

9248  9253  9258  9263 

9269 

9274  9279  9284  9289 

11 

10 

9 

8 

85 

9294 

9299  9304  9309  9315 

9320 

9325  9330  9335  9340 

5 

1 

1.1 

1.0 

0.9 

0.8 

2 

2.2 

2.0 

1.8 

1.6 

86 

9345 

9350  9355  9360  9365 

9370 

9375  9380  9385  9390 

3 

3.3 

3.0 

2.7 

2.4 

87 

9395 

9400  9405  9410  9415 

9420 

9425  9430  9435  9440 

4 

4.4 

4.0 

3.6 

3.2 

88 

9445 

9450  9455  9460  9465 

9469 

9474  9479  9484  9489 

5 

5.5 

5.0 

4.5 

4.0 

89 

9494 

9499  9504  9509  9513 

9518 

9523  9528  9533  9538 

(i 

6.6 

6.0 

5.4 

4.8 

7 

7.7 

7.0 

6.3 

5.6 

90 

9542 

9547  9552  9557  9562 

9566 

9571  9576  9581  9586 

4 

8 

8.8 

8.0 

7.2 

6.4 

9 

9.9 

9.0 

8.1 

7.2 

91 

9590 

9595  9600  9605  9609 

9614 

9619  9624  9628  9633 

92 

9638 

9643  9647  9652  9657 

9661 

9666  9671  9675  9680 

93 

9685 

9689  9694  9699  9703 

9708 

9713  9717  9722  9727 

7 

6 

5 

4 

94 

9731 

9736  9741  9745  9750 

9754 

9759  9763  9768  9773 

1 

0.7 

0.6 

0.5 

0.4 

95 

9777 

9782  9786  9791  9795 

9800 

9805  9809  9814  9818 

5 

2 

1.4 

1.2 

1.0 

0.8 

3 

2.1 

1.8 

1.5 

1.2 

96 

9823 

9827  9832  9836  9841 

9845 

9850  9854  9859  9863 

4 

2.8 

2.4 

2.0 

1.6 

97 

9868 

9872  9877  9881  9886 

9890 

9894  9899  9903  9908 

5 

3.5 

3.0 

2.5 

2.0 

98 

9932 

9917  9921  9926  9930 

9934 

9939  9943  9948  9952 

(j 

4.2 

3.6 

3.0 

2.4 

99 

9956 

9961  9965  9969  9974 

9978 

9983  9987  9991  9996 

7 

4.9 

4.2 

3.5 

2.8 

8 

5.6 

4.8 

4.0 

3.2 

100 

0000 

0004  0009  0013  0017 

0022 

0026  0030  0035  0039 

4 

9 

6.3 

5.4 

4.5 

3.6 

144 


FIVE-PLACE  LOGARITHMS 


0     1 

234 

567 

8     9 

43 

100  00000  00043 

00087  00130  00173 

00217  00260  00303 

00346  00389 

101 
102 
103 
104 

j  00432  00475 
00860  00903 
01284  01326 
01703  01745 

00518  00561  00604 
00945  00988  01030 
01368  01410  01452 
01787  01828  01870 

00647  00689  00732 
01072  01115  01157 
01494  0153(5  0157S 
01912  01953  01995 

00775  00X17 
01199  01242 
01620  01(502 
02036  02078 

K; 
42 
41 
41 

105 

02119  02160 

02202  02243  02284 

02325  02366  02407 

02449  02490 

41 

106 
107 
108 
109 

02531  02572 
02938  02979 
03342  033X3 
03743  03782 

02612  02653  02694 
03019  03060  03100 
03423  03463  03503 
03822  03862  03902 

02735  02776  02816 
03141  03181  03222 
03543  03583  03(523 
03941  03981  01021 

02X57  02898 
032(52  03302 
030(53  03703 
040(50  04100 

40 
40 
40 
39 

11 
12 
13 
14 

15 

04139  04532 
07918  08279 
•11394  11727 
14613  14922 

17609  17898 

390 
358 
330 
307 

287 

04922  05308  05690 
08636  08991  09342 
12057  123S5  12710 
15229  15534  15836 

18184  18469  18752 

37!) 
349 
323 
300 

281 

06070  0(544(5  06819 
09691  10037  10380 
13033  13351  13(572 
16137  16435  16732 

19033  19312  19590 

370 
341 
316 
294 

276 

07188  07."):).") 
10721  11059 
13988  14301 
17026  17319 

19866  20140 

3(53 
335 
312 
280 

272 

16 
17 
18 
19 

20412  20683 
23045  23300 
25527  25768 
27875  28103 

269 
253 
239 
227 

20952  21219  21484 
23553  23X0.-,  24055 
26007  2(5245  204X2 
2S330  28556  28780 

264 
249 
235 
223 

21748  22011  22272 
24304  24551  24797 
26717  26951  27184 
29003  29226  20447 

259 
245 
232 
220 

22531  22789 
25042  252S5 
27416  27646 
29667  29885 

256 
242 

229 

218 

20 

30103  30320 

216 

30535  30750-30963 

213 

31175  31387  31597 

209 

31806  32015 

207 

21 
22 
23 
24 

32222  32428 
34242  34439 
36173  36361 
38021  38202 

205 
196 

188 
180 

32634  32838  33041 
34635  34830  35025 
36549  3(1736  36922 
38382  38561  38739 

202 
194 
185 
178 

33244  33445  33646 
352  IS  35411  35603 
37107  372'.)  1  3717.") 
38917  39094  39270 

200 
191 
183 
175 

33846  34044 
35793  359.x  1 
37658  37X40 
39445  39620 

198 
1X9 
181 
174 

25 

39794  39967 

173 

40140  40312  40483 

171 

40654  40824  40993 

169 

41162  41330 

167 

26 

27 
28 
29- 

41497  41664 
43136  43297 
44716  44871 
46240  46389 

41830  41996  42160 
43457  43616  43775 
1  :>(>•_'.",  45179  45332 
46538  46687  46835 

163 

1:58 

152 

147 

42325  42488  42651 
'43933  44091  44248 
45484  45637  45788 
46982  47129  47276 

42X13  42975 
44404  44560 
45939  46090 
47422  47567 

161 
156 
160 

I  if, 

30 

47712  47857 

48001  48144  48287 

143 

48430  48572  48714 

48855  4X990 

140 

31 
32 
33 
34 

49136  49276 
50515  50651 
51851  51983 
53148  53275 

49415  49554  49693 
50786  50920  51055 
52114  52244  52375 
53403  53529  53656 

138 
133 
129 
126 

49831  49969  50106 
•Miss  ;,1322  51455 
52504  52(534  527(53 
53782  53908  54033 

.",0243  50379 
515X7  51720 
52X«»2  53020 
54158  54283 

136 
131 
128 
124 

35 

54407  54531 

54654  54777  54900 

123 

55023  55145  552(57 

55388  55509 

121 

36 
37 
38 
39 

40 

55630  55751 
56X20  56937 
57978  58092 
59106  59218 

60206  60314 

55871  55991  56110 
57054  57171  57287 
58206  58320  58433 
59329  59439  59550- 

60423  60531  6063X 

56229  5(5348  56467 
57403  57519  57634 
5S5H5  5X059  5X771 
59000  59770  59879 

60716  00X53  60959 

f,6.-,x5  56703 
.".771!)  57864 
.-,XXS3  .-)X99:, 
599XX  60097 

61066  (51172 

117 
114 

109 
106  ! 

41 
42 
43 

44 

45 

61278  013X4 
62325  62428 
63347  6344N 
64345  64444 

65321  <>.">l  IN 

61490  01  595  (H700 
62531  021)34  62737 
63.">4X  I5364!)  (53749 
64542  64(540  6473S 

65514  65(il()  05700 

(lixor,  01909  02014 
•  52X39  02941  03043 
63849  0394!)  0404.x 
04X30  04933  05031 

65801  G5X90  05992 

| 

621  IX  62221 
(53144  (53240 
(54147  6121(5 
65128  (55225 

660X7  661X1 

104 

101 
99 
96 

95 

46 
47 
48 
49 

66276  66370 
67210  (57302 
68124  68215 
69020  (J9IOS 

664(54  66558  6(5652 
67394  074X0  0757X 
68305  (5X395  t;s4X5 
(59197  69285  (59373 

(50745  66X39  66932 
(570(59  07701  07X52 
68574  (58(564  6X753 
09401  0954X  09030 

0702.")  671  17 
07943  0X034 
OXS42  6X931 
69723  09X10 

93 

90 

88 
87 

no  69X97  699xi 

70070  70157  70243 

7032!)  70415  70501 

f  ^jp 

705X0  70072   X5 

FIVE-PLACE  LOGARITHMS 


145 


0     1 

234       5.67 

8    9 

50 

69897  69984 

70070  70157  70243    70329  70415  70501 

70586  70672 

85 

51 

52 
53 

f,l 

70757  70842 
71600  71684 

72428  72:>()9 
73239  73320 

70927  71012  71096    71181  71265  71349 
71767  71850  71933    72016  72099  72181 
72591  72673  72754    72S35  72916  72997 
73400  73480  73560    73640  73719  73799 

71433  71517 
72263  72346 
73078  73159 
73878  73957 

83 
82 
81 
79 

55 

74036  74115 

74194  74273  74351 

74429  74507  74586 

74663  74741 

78  1 

60 
57 

58 
59 

74819  74896 
75587  75664 
76343  764  IS 
77085  77159 

74974  75051  75128 
75740  75815  75891 
7(1492  76567  76611 
77232  77305  77379 

75205  75282  75358 
7f>907  76042  76118 
76716  76790  76864 
77452  77525  77597 

75435  75511 
76193  76268 
76938  77012 
77670  77743 

76 
75 
73 

72 

60 

77815  77887 

77960  78032  78104 

78176  78247  78319 

78390  78462 

71 

61 
62 
63 
«4 

78533  78604 
79239  79309 
79934  80003 
80618  80686 

78675  78746  78817 
79379  79449  79518 
80072  80140  80209 
80754  80821  80889 

78888  78958  79029 
79588  79657  79727 
80277  80346  80414 
80956  81023  81090 

79099  79169 
79796  79865 
80482  80550 
81158  81224 

70 
69 
68 
67 

(M    81291  81358 

81425  81491  81558 

81624  81690  81757 

81823  81889 

65 

Q6 
67 
68 
69 

81954  82020 
82607  82672 
83251  83315 
83885  83948 

82086  82151  82217 
82737  82802  82866 
83378  83442  83506 
84011  84073  84136 

82282  82347  82413 
82930  82995  83059 
83569  83632  83696 
84198  84261  84323 

82478  82543 
83123  83187 
83759  83822 
84386  84448 

64 
64 
63 

62 

70 

84510  84572 

84634  84696  84757 

84819  84880  84942 

85003  85065 

61 

71 
72 
73 

74 

85126  85187 
85733  85794 
86332  86392 
86923  86982 

85248  85309  85370 
S5.S51  85914  85974 
86451  86510  86570 
87040  87099  87157 

85431  85491  85552 
86034  86094  86153 
86629  86688  86747 
87216  87274  87332 

85612  85673 
86213  86273 

S6XOO  86864 
87390  87448 

60 
59 
59 

58 

75 

87506  87564 

87622  87679  87737 

87795  87852  87910 

87967  88024 

57 

76 

77 
7S 
79 

88081  88138 
88049  88705 
89209  89265 
89703  89818 

88195  88252  88309 
88762  88818  88874 
89321  89376  89432 
89873  89927  89982 

88366  88423  88480 
88930  88986  89042 
89487  89542  89597 
90037  90091  90146 

88538  88593 
89098  89154 
89653  89708 
90200  90255 

56 
55 
55 
54 

80 

90309  90363 

90417  90472  90526 

90580  90634  90687 

90741  90795 

54 

81 
82 
83 
84 

90849  90902 
91381  91434 
91908  91960 
92428  92480 

90956  91009  91062 
91487  91540  91593 
92012  92065  92117 
92531  92583  92634 

91116  91169  91222 
91645  91698  91751 
92169  92221  92273 
92686  92737  92788 

91275  91328 
91803  91855 
92324  92376 
92840  92891 

53 
53 
52 
51 

85 

92942  92993 

93044  93095  93146 

93197  93247  93298 

93349  93399 

51 

86 
87 
88 
89 

90 

93450-93500 
93952  94002 
94448  94498 
94939  94988 

95424  95472 

93551  93601  93651 
94052  94101  94151 
94547  94596  94645 
95036  95085  95134 

95521  95569  95617 

93702  93752.93802 
94201  94250  94300 
94694  94743  94792 
95182  95231  95279 

95665  95713  95761 

93852  93902 
94349  94399 
94841  94890 
95328  95376 

95809  95856 

50 
49 
49 

48 

48 

91 
92 
93 
94 

95904  95952 
96379  96426 
96848  96895 
•  97313  97359 

95999  96047  96095 
96473  96520  96567 
96942  96988  97035 
97405  97451  97497 

96142  96190  96237 
96614  96661  96708 
97081  97128  97174 
97543  97589  97635 

96284  96332 
96755  96802 
97220  97267 
97681  97727 

47 
46 
46 
45 

95 

97772  97818 

97864  97909  97955    98000  98046  98091 

98137  98182 

45 

96    98227  98272    9831  8  98363  98408 
97    9SG77  98722    98707  9S811  98856 
9X    99123  99167    99211  99255  99300 
99    99564  99607    99(151  «.)%«>:•  <»9739 

98453  98498  98543 
98900  98945  98989 
99344  99388  99432 
99782  99820  99870 

98588  98632 
99034  99078 
99476  99520 

99913  99957 

45  ! 
45 

43 

100 

00000  00043    00087  00130  00173 

00217  00260  00303 

00346  003X9 

43 

01       234       567 

8    9 

INDEX 


ABRIDGED    multiplication    and 

division,  17,  43 
Abscissa,  66 

Accidental  errors,  85,  86 
Accuracy,  34,  97 

absolute,  37 

of  average,  105 

decimal,  38 

relative,  38,  41 
Angles/27 

circular  measure,  28,  53 

numerical  value,  28,  53 
Antilogarithm,  46 
Approximate  check,  57 
Area 

unit  of,  22 

measurement  of,  25 
Arithmetical  mean,  93 
Average,  93,  123 

advantage  of,  97,  115 

deviation,  98,  101,  123 

by  partition,  98 

by  symmetry,  98 
Axes  of  x  and  y,  62 


B 


BALANCE,  26 
Base,  logarithmic,  45 
Black  thread  method,  70 
in  non-linear  cases,  73,  74 


C.  G.  S.  system,  20 
Caliper  (see  Vernier). 


Characteristic 

of  a  logarithm,  47 

deviations,  100,  123 

magnitudes,  93 
Chauvenet's  criterion,  116,  119, 

120 

Checking  by  approximation,  57 
Circular 

functions,  33,  61 

measure,  28,  53 
Class  interval,  89 
Coincidence, 

principle  of,  77 

measurement  by,  79 
Cologarithm,  48 
Common  logarithms,  45 
Conditioned  measurements,  82, 

124 

Constant  errors,  85,  86,  134 
Cosine,  33 

Criteria  of  rejection,  115 
Curve 

of  a  +  bx,  68 

of  a  +  bx  +  ex*,  71 

of  e-*2,  67,  92,  117 

of  an  equation,  65 

of  log  x,  67,  75 

of  x/a  +  y/b  =  1,  71 
Cylinder  gauge  points,  61 


D 

DELTAS,  49,  76 
Density,  22 

measurement  of,  26,  110 

of  water,  72 
Dependent 

measurements,  82 

variable,  64 


148 


INDEX 


Deviation,  96 

average,  98,  101,  123 

characteristic,  101,  123 

and  dispersion,  100 

of  measurements,  83 

standard,  102 
Diagonal  scale,  29 
Direct  measurements,  82 
Dispersion,  102,  123 

of  average,  107 

relative,  109 

and  semi-interquartile  range, 
103,  104 

significance  of,  103 

of  single  measurements,   107 

with  slide  rule,  106 

and  weighting,  113 

double  weighing,  53 


Frequency,  types,  90 
Frequency  polygon,  89 

area  of,  99,  117 
Function,  33 


G 

GEOMETRICAL  mean,  93 
Graph  (see  curve). 
Graphic 

analysis,  68 

criteria  of  rejection,  120 

diagram,  62 

of  an  equation,  65 

extrapolation,  77 

interpolation,  75 

representation,  62 


e,  45 
e-x2 

calculation  of,  48 

curve  of,  67,  92,  117 
Equation 

curve  of,  65 

graph  of,  65 

of  a  parabola,  67,  71 

of  a  straight  line,  69 
Equipment,  13 
Equivalent  measures,  59 
Equivalents,  slide  rule,  81 
Errors,  85 

constant,  134 

and  deviations,  97 

periodic,  130 

progressive,  129,  133 

systematic,  129 
Estimation 

of  length,  36,  78 

of  tenths,  34 
Extrapolation,  graphic,  77 


H 

HALF  (see  Rounding  off). 
Hand,  for  measuring,  24 
Harmonic  mean,  93 
Histogram,  89 

area  of,  99 
Honesty  of  observation,  115 


I 


INCH,  length  of,  40 
: Independent 

measurements,  82 

variable,  63 
Indirect  measurements,  82 

probable  errors  of,  126 
Intercepts,  69 
interpolation 

graphic,  75 

tabular,  47 
Interquartile  range,  99 


F 

FIGURES,  significant,  34 
Frequency  distribution,  89 


LEAST  squares,  123 
Length 


INDEX 


149 


Length,  by  estimation,  36 

measurement,  23 

units  of,  20 
Line,  equation  of,  69 
Linear  law,  69 

reduction  to,  74 
Log  x,  curve  of,  67,  75 
Logarithms,  45,  61 
Lower  quartile,  99 

M 

MASS 

by  double  weighing,  53 

measurement  of,  26 

units  of,  22 
Mean 

arithmetical,  93 

choice  of,  94 

geometrical,  93 

harmonic,  93 

quadratic,  93 

Means,  relations  between,  94 
Measurements 

C.  G.  S.,  23 

classification  of,  82 

by  coincidence,  79 

and  errors,  82 

by  estimation,  78 

physical,  37 

statement  of,  108 

weighting  of,  109 
Median,  93,  99 

advantages  of,  95,  115 
Metric  system,  20 
Mode,  93,  94 


X 


NATURAL  logarithms,  45 
Negative  slope,  69 
Negligible  magnitudes,  49 
Numerical  value   of   an   angle, 
*  28,  53 

O 

OBSERVATIONAL  honesty,  115 
Observations,  weighting  of,  109 
Ordinate,  66 


Origin 

change  of,  73 

of  graphic  diagram,  69 


rr,  37 

experimental     determination 

of,  37 

Parabola,  67,  71 
Periodic  errors,  130 
Physical  measurement,  37 
Prejudice,  82,  113,  116 
Principle  of  coincidence,  77 
Probability  curve,  67,  92,  117 
Probable  error,    119,    123    (see 

dispersion), 
of  indirect  measurements, 

126 

Progressive  errors,  129,  133 
Proportion,  14 
Proportional 

accuracy,  41 

difference,  42 

errors,  41 

Prototype  standards,  21 
Protractor,  28 

Q 

Quadratic  mean,  93 
Quartiles,  99 

R 

RANGE 

interquartile,  99 

semi-interquartile,  99 

total,  101 

Rejection,  criteria  of,  115 
Relative 

accuracy,  41 

difference,  42 

errors,  41 

of  sines,  33 
Representative 

deviations,  100 

magnitudes,  93 
Rounding  off,  23,  39,  90 


150 


INDEX 


2,  100 

Scales  of  a  graph,  60 

Semi-quartile    range,    99,    103, 

104 

Sigma  notation,  106 
Significant 

figures,  34,  39 

zero,  39 
Sine,  32,  61 
Slide  rule,  54 

for  dispersion,  106 

equivalents,  81 
Slope,  30 

negative,  69 

Small  magnitudes,  49,  76 
Smoothed  curves,  65 
Square  roots,  60 
Squares,  60 
Standard 

deviation,  102,  123 

form,  44 

Standards  of  measurement,  20 
Statement   of   a  measurement, 

108 

Statistics,  89 
Straight  line 

equation  of,  69 

law,  69 

reduction  to,  74 
Systematic  errors,  129 


TABLES 

circular  functions,  141 
logarithms,  142-145 
probable  errors,  141 
sines,  141 
squares,  141 
tangents,  141 
use  of,  31,  47,  105 

Tangent,  30,  61 

Temperature,  daily,  64 

Tenths,  estimation  of,  34 


Time,  unit  of,  22 
Total  range,  101 


U 

UNITS  of  measurement,  20 
Upper  quartile,  99 


VARIABLE 

dependent,  64 

independent,  63 
Variates,  88 

Variation  and  proportion,  14 
Variations,  87 
Vernier  caliper,  79,  88 
Volume 

measurement  of,  25 

unit  of,  22 


W 

WATER,  density  of,  72 
Weighted  average,  112 
Weighting  of  observations,  109 
Weights 

and  dispersion,  113 

and  measures,  20 
Wright's  criterion,  122 


X 


x-axis,  62 


is,  63 


ZERO,  significant,  39 


FOUR-PLACE  AND  FIVE-PLACE  TABLES 

From  Turtle's  Introduction  to  Laboratory  Physics 
Cardboard,  6  pages.    4|  X  7^  inches.    Price,  5  cts. 

A  four-place  logarithm  table  arranged  as  carefully 
as  the  larger  tables  used  by  mathematicians  and 
astronomers.  The  columns  are  marked  off  in  the  way 
that  will  make  their  use  most  convenient  to  physicists 
and  others  who  are  accustomed  to  using  scales  that  are 
subdivided  into  tenths. 

A  two-page,  five-place  logarithm  table  that  can 
be  relied  upon  is  included  for  occasional  use  when 
five-figure  calculations  are  needed. 

A  four-place  table  of  radian  measures,  natural  sines, 
cosines,  and  tangents,  and  logarithmic  sines  and 
cosines,  for  each  degree  of  the  quadrant,  is  arranged 
with  the  logarithmic  columns  side  by  side  so  as  to 
facilitate  taking  out  logarithmic  tangents  by  mental 
subtraction. 

A  short  list  of  mathematical  constants  and  their 
logarithms  is  included;  also  a  table  of  squares,  and  a 
table  of  two-figure  values  of  .67449  ^n  for  all  five- 
figure  values  of  n. 


AN  INTRODUCTION  TO  LABORATORY  PHYSICS 

By  Lucius  Turtle,  M.D.,  Associate  in  Physics,  Jefferson  Medi- 
cal College.   Cloth,  150  pages.    5  X  7 !  inches.    Price,  80  cts. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


WAR  10  ^21 

3 

^jai^BP 

\\ 

m  «^ 

...  o 

(k  (M 

$OCV54BH 

I8orfw*c 

REC'D  LD 

OCT15'64-4Pf 

t 

LD  21-100m-7,'40  (6936s) 

YB  09801 


673213 


WWER  DIVISION 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


